Annals of Mathematics, 151 (2000), 625–704
The monodromy groups of Schwarzian
equations on closed Riemann surfaces
By Daniel Gallo, Michael Kapovich, and Albert Marden
To the memory of Lars V. Ahlfors
Abstract
Let θ : π1 (R) → PSL(2, C) be a homomorphism of the fundamental group
of an oriented, closed surface R of genus exceeding one. We will establish the
following theorem.
Theorem. Necessary and sufficient for θ to be the monodromy representation associated with a complex projective stucture on R, either unbranched
or with a single branch point of order 2, is that θ(π1 (R)) be nonelementary.
A branch point is required if and only if the representation θ does not lift to
SL(2, C).
Contents
1. Introduction and background
2. Fixed points of Möbius transformations
A. The pants decomposition
3. Finding a handle
4. Cutting the handles
5. The pants decomposition
B. Pants configurations from Schottky groups
6. Joining overlapping plane regions
7. Pants within rank-two Schottky groups
8. Building the pants configuration
9. Attaching branched disks to pants
10. The obstructions
C. Ramifications
11. Holomorphic bundles over Riemann surfaces, the 2nd Stiefel-Whitney
class, and branched complex projective structures
12. Open questions about complex projective structures
References
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DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
1. Introduction and background
1.1. Introduction. The goal of this paper is to present a complete, selfcontained proof of the following result:
Theorem 1.1.1. Let R be an oriented closed surface1 of genus exceeding
one, and
θ : π1 (R; O) → Γ ⊂ PSL(2, C)
a homomorphism of its fundamental group onto a nonelementary group Γ of
Möbius transformations. Then:
(i) θ is induced by a complex projective structure for some complex structure
on R if and only if θ lifts to a homomorphism
θ∗ : π1 (R; O) → SL(2, C).
(ii) θ is induced by a branched complex projective structure with a single
branch point of order two for some complex structure on R if and only if
θ does not lift to a homomorphism into SL(2, C).
The terms will be explained in §§1.2–1.4.
Theorem 1.1.1 characterizes the class of groups arising as monodromy
groups of Schwarzian differential equations or equivalently, of the projectivized
monodromy groups for the associated linear second-order differential equations. Poincaré himself explicitly raised the question by noting (for punctured
spheres) second-order equations depend on the same number of parameters
as their monodromy groups (the position of the singularities–the conformal
structure–is allowed to change) and from this observation boldly concluded,
“On peut en général trouver une équation du 2d ordre, sans points à apparence singulière qui admette un groupe donné” [P, p. 218]. In our own time,
the question was raised in [Gu3] and [He1]; in fact Gunning conjectured Part
(i) of our theorem and Tan [Tan] conjectured Part (ii).
Schwarzian equations themselves have long been an important tool in the
study of Riemann surfaces and their uniformization. Their relation with algebraic geometry was established by Gunning in [Gu1]: For a fixed complex
structure on R, the linear monodromy representations of the complex projective structures correspond to flat maximally unstable rank 2 holomorphic
vector bundles over R. A similar relation for branched structures was later
studied by Mandelbaum; see e.g. [Man 2, 3] (see also §11).
1 In
this paper, all surfaces are assumed to be connected. A closed surface is one which is compact,
without boundary.
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627
In §11, we will present an analogue Theorem 11.3.3 of our main theorem in
the context of holomorphic vector bundles over Riemann surfaces. Namely, let
S be an oriented closed surface of genus exceeding one and ρ : π1 (S) → SL(2, C)
a nonelementary representation. Then ρ is the monodromy of a holomorphic
flat connection on a maximally unstable holomorphic vector bundle of rank two
over a Riemann surface R, where R is diffeomorphic to S via an orientation
preserving diffeomorphism R → S.
Besides the fuchsian groups of uniformization, the class of monodromy
groups includes the discrete, isomorphic groups of quasifuchsian deformations
(Bers slices which model Teichmüller spaces and their boundaries), and discrete
groups such as Schottky groups which are covered by fuchsian surface groups.
See [Mas2] for a wide array of possibilities.
Theorem 1.1.1 further implies that the image in PSL(2, C) of “almost”
every homomorphism of the fundamental group has a geometric structure. This
is quite astonishing, especially so as the image groups are often not discrete
and not even finitely presentable.
R. Rubinsztein [R] observed that if G0 ⊂ G = π1 (R) is any index two
subgroup, the restriction of θ to G0 can be lifted from PSL(2, C) to SL(2, C)
in 22g ways. Consequently by Theorem 1.1.1, a homomorphism whose restriction to an index two subgroup is nonelementary is always associated with a
complex projective structure for some complex structure on the corresponding
two sheeted cover. One such index two subgroup is constructed in §8.6.
Special cases of Theorem 1.1.1(i) were proved in [He1] and the case of
homomorphisms into PSL(2, R) was investigated in [Ga], [Go], [Por] and [Tan].
Proofs of Theorem 1.1.1(i) have been announced before. Gallo’s research
announcement [Ga1] proposed an innovative strategy for a proof, but the
promised details have not been published or confirmed. Gallo’s strategy had
been developed in consultation with W. Goldman and W. P. Thurston, and
was particularly inspired by Thurston’s approach to the deformation of fuchsian groups by bending. Goldman’s paper [Go1] is an exemplar of this strategy
applied in the interesting special case where θ is an isomorphism onto a fuchsian group; it deals with the problem of determining all complex projective
structures with the prescribed monodromy. This question is discussed further
in §12.
The recent paper [Ka] proposed a proof confirming Theorem 1.1.1(i). Although the argument presented is incomplete (Lemmas 1 is incorrect and a
condition is omitted in Lemma 4; they are corrected in the present paper,
and some details are missing in the proofs of Propositions 1 and 2), the paper
contains new ideas and directly motivated a fresh examination of the whole
issue.
The present work was begun by Marden with the goal of settling the validity of the claims. In a general sense, Gallo’s and Kapovich’s strategy is
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DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
followed, although the details, especially in Part B, are quite different from
those suggested in [Ga1] or [Ka]. In the latter phase of the investigation,
a collaboration with Kapovich began. Almost immediately this produced a
breakthrough in understanding the connection between a certain construction
invariant and the lifting obstruction (§§9–10). Instead of using the difficult
continuity arguments proposed in [Ka], we use branched structures. Motivated by Tan’s work [Tan] on real branched structures, we found a technique
for constructing branched projective structures complementing that developed
earlier for joining pants. This approach exhibits clearly the connection. It also
clarifies the role of the 2nd Stiefel-Whitney class and degree of instability of
holomorphic bundles which is discussed by Kapovich in §11. In fact, one of
our discoveries is that it is easier to prove Theorem 1.1.1 simultaneously for
branched and unbranched structures than to establish the unbranched case by
itself.
Part C of our paper brings together additional results that fill out the
picture presented by our main theorem. These are developed in the context of holomorphic bundles over Riemann surfaces. For example, in some
respects Theorem 1.1.1 is more clearly seen in the context of a more general existence theorem for branched complex projective structures with a prescribed branching divisor and monodromy representation. This refinement,
Theorem 11.2.4, is expressed in terms of the 2nd Stiefel-Whitney class. In
addition, we present the full proof of the divergence theorem briefly outlined
in [Ka]. This Theorem 11.4.1 deals with sequences of monodromy homomorphisms θn : π1 (R) → PSL(2, C) associated with divergent Schwarzian equations on a fixed Riemann surface. Such a sequence of homomorphisms cannot
converge algebraically to a homomorphism, either nonelementary or elementary. In terminology of Teichmüller theory, the extension of a Bers slice to
the full representation variety is properly embedded. In §12 we list and briefly
discuss a number of open problems arising from our work.
We three authors decided to join together to pool the fruits of a decade
of our individual and collaborative research relating to the main result. By
doing so we have arrived at a rather larger understanding of the fundamental
existence problem for the monodromy of projective structures.
Our topic falls under the ancient and revered subject heading of linear
ordinary differential equations on Riemann surfaces, a subject introduced by
Poincaré. The problem we consider fits comfortably with those associated
with “the Riemann-Hilbert Problem” (Hilbert’s 21st problem) for first-order
fuchsian systems and n-th order fuchsian equations. Yet our approach is quite
different than that associated with this theory [A-B], [I-K-Sh-Y], [Sib], [Y],
[He2]. For one thing, our approach is special to second order equations. Then
we work primarily with projectivized monodromy in PSL(2, C). This turns the
problem into one largely involving the geometry of surfaces and Möbius groups.
629
MONODROMY GROUPS
Another difference is that here we are mainly dealing with equations without
singularities. Finally we do not prescribe the complex structure in advance,
rather it is determined as part of the solution: the number of parameters in the
equations matches the number in the representations. The need to introduce
a branch point to handle part (ii) of our theorem is however reminiscent of the
need for “apparent singularities” in that theory.
Except for a particular case, we have left aside the general existence problem for surfaces with punctures and branch points. However, we believe that
the foundation laid here will stimulate (further) exploration of these and other
important aspects of the subject, including a characterization of the nonuniqueness, that are not now well understood.
Acknowledgments. Marden would like to thank the Mathematics Institute
of the University of Warwick, the Forschungsinstitut für Mathematik at ETH,
Zürich, and the Mathematical Sciences Research Institute in Berkeley, for the
privilege of participating in their programs while his research was carried out.
In addition he thanks David Epstein, Dennis Hejhal, Yasutaka Sibuya, and
Kurt Strebel for helpful discussions. David in particular provided insightful
suggestions for some of the proofs.
This research additionally received support from the NSF grants DMS9306140 and DMS-96-26633 (Kapovich) and DMS-9022140 at MSRI (Kapovich
and Marden).
All us authors thank Silvio Levy for providing invaluable editorial and
a
L TEX assistance and the referee for many helpful comments and suggestions.
1.2. Möbius transformations. Möbius transformations correspond to elements of PSL(2, C) according to
α(z) =
az + b
cz + d
µ
←→
±
a
c
b
d
¶
with ad − bc = 1.
They extend from their action on the extended plane C ∪∞ to upper half-threespace or, via stereographic projection, from the 2-sphere S2 to the
3-ball. The extensions form the group of orientation-preserving isometries
of hyperbolic three-space, which we denote by H3 (in either the ball model or
the upper half-space model) with ∂ H3 denoting the “sphere at infinity,” that
is, the extended plane or S2 , depending on the model. Throughout our paper,
we will identify the extended plane with S2 .
We recall the standard classification:
• A transformation α is parabolic if it has exactly one fixed point on ∂ H3 ,
or, equivalently, if it is not the identity and its trace satisfies tr2 α =
(a + d)2 = 4. Parabolic transformations are those conjugate to z 7→ z + 1.
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DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
• An elliptic transformation has two fixed points in ∂ H3 and also fixes
pointwise its axis of rotation, that is, the hyperbolic line in H3 joining
the fixed points. Its trace satisfies 0 ≤ tr2 α < 4, and it is conjugate to
an element of the form z 7→ e2iθ z, for 0 < θ < π.
• A loxodromic transformation α likewise has two fixed points in ∂ H3 , one
repulsive and the other attractive; it preserves the line in H3 between
/ [0, 4], and α
them which is called the axis. The trace of α satisfies tr2 α ∈
is conjugate to z 7→ λ2 z, where λ satisfies |λ| > 1 and tr2 α = (λ + λ−1 )2 .
The transformation α acts on its axis by by moving each point hyperbolic
distance 2 log |λ| toward the attractive fixed point.
The identity is not part of this classification.
A group Γ is elementary if there is a single point on ∂ H3 , or a pair of
points on ∂ H3 , or a single point in H3 , which is invariant under all elements
of Γ.
The generic group Γ with two or more generators is nonelementary, and is
likely to be nondiscrete as well. For example, any two loxodromic transformations α and β without a common fixed point generate a nonelementary group
Γ = hα, βi. The group Γ is the homomorphic image, in many ways, of any
surface group of genus g ≥ 2.
The most important class of groups ruled out by the condition that Γ be
nonelementary are groups of rotations of the two-sphere and groups conjugate
to them (unitary groups). We recall that a group, discrete or not, that is
composed solely of elliptic transformations is conjugate to a group of rotations
of the 2-sphere.
In anticipation of our later work, we also recall the definition of a twogenerator classical Schottky group G = hα, βi. There are four mutually disjoint circles with mutually disjoint interiors, arranged as two pairs (c1 , c01 ) and
(c2 , c02 ). The generator α sends the exterior of c1 onto the interior of c01 , and β
does the same for (c2 , c02 ). The common exterior of all four circles serves as a
fundamental region for its action on its regular set Ω.
Let π : Ω → S := Ω/G denote the natural projection. The surface S has
genus two, and π(c1 ) and π(c2 ) are disjoint, nondividing simple loops on S. If
d ⊂ S is a simple loop with an α-invariant lift d∗ ⊂ Ω, the free homotopy class
of d in S is uniquely determined up to Dehn twists about π(c1 ) (see §1.8).
The group G extends to act on Ω ∪ H3 ; the quotient is a handlebody of
genus two in which π(c1 ) and π(c2 ) are compressing loops that bound mutually
disjoint compressing disks in the interior.
If, instead of circles, the pairs (c1 , c01 ) and (c2 , c02 ) are Jordan curves (which
can always be assumed to be smooth), the resulting group is called more generally a (rank-two) Schottky group. According to [Chu], or [Z] in the handlebody
MONODROMY GROUPS
631
interpretation, every set of free generators of a Schottky group (of the general
kind!) corresponds to pairs of Jordan curves as described above.
Our method of construction in this paper will always yield classical Schottky groups in terms of designated generators. The extra knowledge that, for
the designated generators, the loops can be taken as round circles is pleasing
and convenient, but it is not really necessary for the proofs.
1.3. Projective structures. Let R be a closed Riemann surface of genus at
least two, and let R = H2 /G be its representation in the universal covering
surface H2 (the two-dimensional hyperbolic plane) by a fuchsian covering group
G. We will describe a projective structure first in the universal cover H2 and
then intrinsically in R.
A complex projective structure with respect to G is a meromorphic, locally univalent (i.e. locally injective) function f : H2 → f (H2 ) ⊆ S2 , for
which there corresponds a homomorphism θ : G → Γ ⊂ PSL(2, C) such that
f (γ(t)) = θ(γ)f (t) for any t ∈ H2 and any γ ∈ G. It follows that f descends
to a multivalued function f∗ on R, called the (multivalued ) developing map; it
“unrolls” R onto the sphere. The Schwarzian derivative of f ,
µ
(1)
St (f ) :=
f 00
f0
¶0
µ
−
1 f 00
2 f0
¶2
= φ(t),
satisfies φ(γ(t))γ 0 2 (t) = φ(t), and therefore descends to a holomorphic quadratic
differential on R.
Conversely, given any holomorphic φ(t) in H2 with this invariance under
G, there is a solution f (t) of (1), uniquely determined up to post composition
by Möbius transformations, which is a locally univalent meromorphic function
that induces a homomorphism θ of G.
The Schwarzian equation is related to the second-order linear differential
equation
(2)
u00 (t) + 12 φ(t)u(t) = 0
as follows. The ratio f (t) = u1 (t)/u2 (t) of any two linearly independent solutions u1 and u2 in H2 gives a solution f of the Schwarzian; conversely, any
solution f of the Schwarzian can be so expressed, indeed
(3)
1
u2 = (f 0 )− 2 ,
u1 = f u2 ,
if the Wronskian ∆(u1 , u2 ), which is necessarily a constant, is normalized as
∆ = 1. Another pair au1 + bu2 , cu1 + du2 of independent solutions corresponds
to the solution Bf of the Schwarzian, where B(z) = (az + b)/(cz + d).
On the Riemann surface R = H2 /G, a form of (2) that is invariant under
change of local coordinates z is,
(4)
v 00 (z) + 12 {φ(z) + Sz (π −1 )}v(z) = 0,
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DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
where π denotes the projection from H2 . In interpreting this equation, the
Schwarzian transforms as a connection under change of local coordinate z 7→
ζ = ζ(z) and v transforms as a half-order differential (see [Ha-Sch]), specifically
1
−2
v(ζ(z))ζ 0 (z)
= v(z).
The monodromy group and monodromy representation are computed
as follows. Fix π1 (R; O) with basepoint O ∈ R, and a solution f∗ (z) (or
v1 (z)/v2 (z)) near O. Let c ∈ π1 (R; O) be a simple loop based at O. Analytically continue f∗ (or v1 /v2 ) around c, arriving back at a solution γf∗ (or
γ(v1 /v2 )), for γ ∈ PSL(2, C). Set θ(c) = γ. In this manner the local solutions
f∗ (or v1 /v2 ) determine a monodromy epimorphism
θ : π1 (R; O) → Γ ⊂ PSL(2, C),
where Γ is a monodromy group for the equation. A different local solution
Bf∗ (or B(v1 /v2 )), coming possibly from a different choice of basepoint, determines a conjugate homomorphism c 7→ Bθ(c)B −1 . Thus, the equation itself
determines a conjugacy class of homomorphisms into PSL(2, C).
If P is a fundamental polygon for G in H2 , we can regard f (P) as spread
over the Riemann sphere, a membrane in Hejhal’s terminology [He1]. The
θ-image of the edge pairing transformations of P will be edge-pairing transformations of the membrane f (P), which therefore serves as an organizing
principle for Γ.
From the topological point of view, a projective structure is defined by an
orientation preserving local homeomorphism, called the (multivalued ) developing map, of R into S2 or, the (single valued ) developing map of the universal
cover R̃ into S2 which is equivariant with respect to the given homeomorphism
θ. From this perspective, the group Γ is called the holonomy (or, more classically, monodromy) group. There is a unique complex structure on R for which
the local homeomorphism becomes conformal.
The fact that the Schwarzian equation can be replaced by the linear differential equation implies the following:
Lemma 1.3.1. If the homomorphism θ : π1 (R; O) → Γ ⊂ PSL(2, C) is
induced by a projective structure on R, it can be lifted to a homomorphism
θ∗ : π1 (R; O) → Γ∗ ⊂ SL(2, C).
Proof. Consider an action of G ∼
= π1 (R, O) on H2 given by the uniformization of the surface R, take an element h ∈ G. Then the solution pair (3) changes
under analytic continuation from t to T = h(t) according to (see [Ha-Sch])
µ √ 0¶
µ
¶µ √ 0 ¶
q
1/ f
a b
1/ f
0
√ 0 (T ) = h (t)
√
(t),
(5)
f/ f
f/ f0
c d
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MONODROMY GROUPS
where
µ
a b
c d
¶
∈ SL(2, C),
θ(h) =
p
az + b
.
cz + d
There are 22g possible choices for h0 (t) over a set of canonical generators {h}
of G. After we make a choice we get the homomorphism
θ : G → SL(2, C),
∗
∗
µ
θ (h) =
a
c
b
d
¶
∈ SL(2, C).
Note however that θ∗ is not canonically determined by the differential equation
(2).
We emphasize that our notion of lifting does not require that the image
Γ of θ be isomorphic to the image of the lift θ∗ . For example, a lift to SL(2, C)
of a half-rotation in PSL(2, C) has order four, not two.
We will refer to θ∗ as a linear monodromy representation of the projective
structure.
Remark 1.3.2. The projective structure associated with the equation
Sz (f ) = φ can be joined to the identity by means of solutions of Sz (f ) = tφ,
for t ∈ C.
1.4. Branched projective structures. A branched projective structure on a
hyperbolic Riemann surface R is a holomorphic mapping f : H2 → S2 which
is locally univalent except in a discrete subset of H2 and which is equivariant
with respect to a homomorphism θ : G → PSL(2, C). We will say that such a
structure is singly branched if f 0 (z) has at most simple zeroes and the projection
of the set {z : f 0 (z) = 0} to R is exactly one point q. These are the structures
which appear in Theorem 1.1.1 and we will restrict our comments here to
this special case. The more general case will be discussed separately in §11.
Near such point q (which we will identify with zero in local coordinates), the
quadratic differential φ = Sz (f ) has a Laurent expansion of the form,
(6)
φ(z) =
∞
b X
−3
+
ai z i ,
+
2z 2 z i=0
b2 + 2a0 = 0.
Conversely, if φ(z) has such an expression near z = 0, a solution of the
Schwarzian will be of the form f (z) = az 2 (1 + o(1)) near z = 0. With φ
given by (6), the equation (2) has the two linearly independent solutions with
expansions near z = 0 of the form
v1 (z)
= z 3/2 (1 + o1 (1)),
v2 (z)
= z −1/2 (1 + o2 (1)).
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DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
A circuit about z = 0 generates the monodromy
µ
u1
u2
¶
µ
¶
µ
u1
7 J
→
,
u2
where
J=
−1
0
¶
0
.
−1
The projectivized monodromy in PSL(2, C) is just the identity.
Therefore the branched structure determines the homomorphism
θ : π1 (R; O) → PSL(2, C)
as in the unbranched case. However, θ cannot be lifted to a homomorphism
into SL(2, C). Indeed, given a standard presentation
ha1 , b1 , . . . , ag , bg |
Q
[bi , ai ] = 1i
for π1 (R; O), and matrix representations Ai and Bi for θ(ai ) and θ(bi ), we have
θ
∗
µY
¶
[bi , ai ] =
Y
[Bi , Ai ] = J,
where θ∗ (ai ) = Ai and θ∗ (bi ) = Bi .
We will discuss this matter further in §§11.5, 11.6.
T
1.5. Parameter count. The vector bundle Qg of quadratic differentials
over Teichmüller space g has complex dimension 6g − 6. Likewise, the representation variety Vg of homomorphisms θ : π1 (R; O) → PSL(2, C), modulo
conjugacy, has complex dimension 6g − 6. Let Vg0 ⊂ Vg denote the subset
of nonelementary representations, i.e. equivalence classes of homomorphisms
whose images are nonelementary subgroups of PSL(2, C). Theorem 1.1.1 asserts that the map Pg of projective structures Qg → Vg is surjective onto the
component of Vg0 consisting of representations liftable to SL(2, C). In fact, the
image space Vg0 is itself a complex analytic manifold [Gu3], [He1]. According
to [Go2], or as a consequence of Theorem 1.1.1, it has two components (one
corresponds to liftable representations and the other one to unliftable representations). See [Ben-C-R] and [Li] for more information about representation
varieties of surface groups.
According to Hejhal’s holonomy theorem [He1] the map Pg is a local homeomorphism which is shown in [E] to be locally biholomorphic. In particular,
the set of points with a given monodromy θ is discrete. According to (1) in
§1.6 below, there is at most one representative in the fiber over a particular
Riemann surface. However Pg is not a covering map [He1].
In Theorem 11.5.2 we will prove an analogue of Hejhal’s holonomy theorem
for singly branched projective structures; we prove that the holonomy mapping
from the space of singly branched projective structures to Vg is locally a fiber
bundle with fiber of complex dimension 1.
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635
1.6. The global structure. Recorded below are basic facts about projective
structures. For the unbranched case, proofs are in [Gu1] and [Kra1, 2]. Other
useful references are [Gu3] and [He1]; the latter includes extensive historical
background.
Here is a brief proof that (in the unbranched case) the holonomy group
Γ = θ(G) cannot be a unitary group, that is, cannot be conjugate to a group of
isometries of S2 . Assume otherwise. Then Γ preserves the spherical metric ρ.
Its pullback f ∗ ρ is a G-invariant metric on H2 which is locally isometric to
the sphere. Consequently f ∗ ρ has constant curvature +1, in violation of the
Gauss-Bonnet theorem.
For the case of a singly branched structures, property (1) below is a special
case of [He1, Theorem 15], (2) will be established as Theorem 11.6.1, and (3)
will be established as Corollary 11.6.1.
Below we consider projective structures σ on R = H2 /G which have the
holomorphic developing mapping f : H2 → S2 and monodromy representation
θ : G → PSL(2, C). Assume that σ is either unbranched (i.e. f is locally univalent) or is singly branched. Let θ(G) = Γ ⊂ PSL(2, C) denote the monodromy
group. The following three properties hold:
(1) If two developing mappings f1 and f2 determine the same homomorphism
θ, then f1 = f2 .
(2) Γ is a nonelementary group.
(3) The following statements are equivalent provided that, when σ is branched,
f (H2 ) is not a round disk in S2 :
(i) f (H2 ) 6= S2 ;
(ii)
(iii)
H2 → f (H2 ) is a possibly branched cover;
Γ acts discontinuously on f (H2 ).
Property (1) does not rule out the possibility that the same target group
Γ may arise from different projective structures on R. Property (2) shows
that the requirement in Theorem 1.1.1 that Γ be nonelementary is necessary.
The situation (3) has a rich structure as it is associated with the theory of
covering surfaces; in particular it includes the theory of quasifuchsian groups
and Schottky groups. In contrast, in the general case there is a bare minimum
of structure because Γ need not be discrete.
1.7. Strategy of the proof. Given a homomorphism
θ : π1 (R; O) → Γ ⊂ PSL(2, C)
such that Γ is nonelementary, the strategy consists of two parts.
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DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
Part A (§§3–5). Find a pants decomposition {Pi } of R with the property
that θ(π1 (Pi )), for 1 ≤ i ≤ 2g −2, is a two-generator (classical) Schottky group.
We recall that a pants is a Riemann surface conformally equivalent to a
three-holed sphere. A surface of genus g ≥ 2 requires 3g − 3 simple loops
to cut it into pants, and there results 2g − 2 pants. It has infinitely many
homotopically distinct pants decompositions.
Part B (§§6–10). Find representations of the universal covers P̃i in the
regular sets (i.e. domains of discontinuity) of θ(π1 (Pi )) ⊂ S2 . Glue them together as dictated by the combinatorics of {P̃i } in R̃, as relayed by θ. In general
there is a Z/2-obstruction to such gluing. If there is no obstruction, we end up
with a simply connected pants configuration S̃ over S2 that models the universal cover of a new Riemann surface S homeomorphic to R. The projection of
S̃ to S2 is a θ-equivariant local homeomorphism. If there is an obstruction, introduce a single branch point of order 2 by applying a twist. This removes the
obstruction to the construction and a new Riemann S surface homeomorphic
to R can be assembled as before. The result is either unbranched or singly
branched projective structure on S with the monodromy representation θ. According to Theorem 11.2.2 if θ lifts to SL(2, C) then the structure has to be
unbranched, if θ does not lift then the structure has to be singly branched; in
other words, the Z/2-obstruction to gluing is the 2nd Stiefel-Whitney class of
the representation θ. This proves Theorem 1.1.1.
The method used to assemble the pants configuration is a form of “grafting,” first applied to kleinian groups in [Mas1].
1.8. Terminology and notation. Throughout this paper we will work on
closed surface R, of genus g ≥ 2. When convenient, we will assume that R is
a Riemann surface R = H2 /G in terms of its universal cover (which may be
taken as the hyperbolic plane H2 ) and fuchsian cover group G. Fix O ∈ R as
the basepoint for its fundamental group π1 (R; O). Let
θ : π1 (R; O) → Γ ⊂ PSL(2, C)
be the designated homomorphism with a nonelementary image Γ.
Throughout we will use lower case Latin letters a, b, c, . . . to denote elements of π1 (R; O), and the corresponding Greek letters α, β, γ, . . . to denote
their θ-images in Γ. A nontrivial loop is one not homotopic to a point.
We will write the compositions of both curves and transformations (and
their associated matrices) starting at the right. Thus, b follows a in both ba
and θ(b)θ(a) = βα.
By a standard set of generators {ai , bi } of π1 (R; O), where 1 ≤ i ≤ 2g,
we mean a set of oriented simple loops that generate the fundamental group
and have the following properties. For each i, the loop bi crosses ai at O, from
MONODROMY GROUPS
637
the right side of ai to the left, and is otherwise disjoint. For j 6= i, the simple
loops (aj , bj ) are freely homotopic to simple loops disjoint from (ai , bi ). The
product of the commutators
Y
−1
b−1
i ai bi ai
i
bounds a simply connected region lying to its left.
We will refer to a product ba as a simple loop if it is homotopic to one
(with fixed basepoint). Thus, for any k ∈ Z, the loop b1 ak1 is simple, and so
−1
k
k
k
are a2 b−1
1 a1 and a2 b1 a1 , but not a2 b1 a1 , or, for k 6= 1, the loop a2 b1 a1 . The
−1
−1
curve b1 a1 b1 is simple, but not a2 b1 a1 b1 .
Often we will modify a simple loop c ⊂ R by applying a Dehn twist,
which can be described as follows. Let A be an annular neighborhood about
a (nontrivial) simple loop a. Orient ∂A so that A lies to its left. Hold one
component of ∂A fixed and rotate the other |n|-times in the positive or negative
direction according to whether n ≥ 1 or n ≤ −1. This action extends to an
orientation preserving homeomorphism δ n of A, and then to all R, by setting
δ n = id outside A. δ n , or more precisely its homotopy class on R, is called the
Dehn twist of order n about a. If c is not freely homotopic to a curve disjoint
from a, then δ n (c) is not freely homotopic to c.
2. Fixed points of Möbius transformations
In this section we will collect the lemmas needed to control the type of
composed transformations.
2.1. Basic lemmas.
Lemma 2.1.1.
(i) Suppose α is loxodromic and β sends neither fixed point of α to the other.
Given M > 0 there exists N ≥ 0 such that |tr βαn | > M and βαn is
loxodromic for all |n| ≥ N .
(ii) Suppose α is loxodromic and β sends exactly one fixed point of α to the
other. Given M > 0 there exists N ≥ 0 such that |tr βαn | > M and βαn
is loxodromic for all n ≥ N (if β sends repulsive to attractive) or for all
n ≤ −N (if β sends attractive to repulsive).
(iii) Suppose α is parabolic and β does not share a fixed point with α. Given
M > 0 there exists N ≥ 0 such that |tr βαn | > M and βαn is loxodromic
for all |n| ≥ N.
638
DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
Proof. For (i) and (ii) we may assume
µ
α=
λ
0
0
λ−1
¶
with |λ| > 1,
µ
β=
a
c
b
d
¶
with ad − bc = 1.
Then tr βαn = λn a + λ−n d. Not both a and d can vanish, because β does not
interchange the fixed points of α. The assertions now follow directly.
For (iii), we may assume
µ
α=
1
0
¶
µ
1
,
1
β=
a
c
b
d
¶
with ad − bc = 1.
Then tr βαn = (a + d) + nc, where c 6= 0. Again, the desired conclusion follows.
Lemma 2.1.2.
Assume α is loxodromic with attractive fixed point p∗
and repulsive fixed point p∗ .
(i) For any sequence k → +∞, the fixed points of βαk converge to β(p∗ ) and
p∗ . The fixed points of αk β converge to p∗ and β −1 (p∗ ).
(ii) For any sequence k → −∞, the fixed points of βαk converge to β(p∗ ) and
p∗ . The fixed points of αk β converge to p∗ and β −1 (p∗ ).
Proof. Part (ii) follows from (i) upon replacing α by α−1 . The computational proof is instructive. Set
µ
α=
λ
0
0
λ−1
¶
µ
and β =
a
c
¶
b
,
d
where |λ| √≥ 1 and ad − bc = 1. If ac 6= 0, the two fixed points of αk β are
λ2k α(1 ± ∆)/2c − d/2c, where
d2
4
2d
+
− 2 2k .
2k
2
4k
aλ
a λ
a λ
The “+” fixed point approaches ∞ with k. The “−” fixed point has the form
∆=1+
√ −1
2
d
d2
d
− −
(1
+
∆) − .
ac c 2acλ2k
2c
This one approaches −b/a = β −1 (0).
If c = 0, one fixed point of αk β is ∞. The other one is b/(dλ−2k − a).
This too approaches −b/a = β −1 (0) with k. √
If a = 0 the two fixed points are (−d ± d2 − 4λ2k )/2c. Both approach
∞ with k. Here β −1 (0) = ∞.
The fixed points of βαk = β(αk β)β converge to β(p∗ ) and β(β −1 p∗ ) = p∗ .
639
MONODROMY GROUPS
Lemma 2.1.3.
Suppose γ is loxodromic with attractive fixed point p∗ ,
repulsive fixed point p∗ .
(i) Suppose α(p∗ ) 6= p∗ and β(p∗ ) 6= p∗ . Given M > 0 there exists N ≥ 0
such that |tr γ −n αγ n β| > M and γ −n αγ n β is loxodromic and does not
share a fixed point with α or β for all n ≥ N .
(ii) Suppose α(p∗ ) 6= p∗ and β(p∗ ) 6= p∗ . Given M > 0 there exists N ≥ 0
such that |tr γ −n αγ n β| > M and γ −n αγ n β is loxodromic and does not
share a fixed point with α or β for all n ≤ −N .
Proof. We may assume
µ
γ=
λ
0
¶
0
,
λ−1
µ
α=
¶
a b
,
c d
µ
β=
u
w
¶
v
,
x
with |λ| > 1, ad − bc = 1, and ux − vw = 1. We find that tr γ −n αγ n β =
λ2n cv + λ−2n bw + (au + dx).
In case (i) we have c 6= 0 (since α(p∗ ) 6= p∗ ), and v 6= 0 (since β(p∗ ) 6= p∗ );
thus cv 6= 0 and γ −n αγ n β is loxodromic for all large n. Moreover, if q is a
fixed point of β, then q 6= p∗ but limn→+∞ γ −n αγ n β(q) = p∗ .
Suppose instead that q is a fixed point of α, and of γ −n αγ n β for all large
n. First q 6= p∗ since γ −n αγ n β(p∗ ) = p∗ implies β(p∗ ) = p∗ . Then β(q) 6= p∗
for γ −n α(p∗ ) = q holds for all large n only if q = p∗ or q = p∗ . Thus once
again, limn→+∞ γ −n αγ n β(q) = p∗ 6= q.
In case (ii), b 6= 0 (since α(p∗ ) 6= p∗ ), and w 6= 0 (since β(p∗ ) 6= p∗ ); hence
−n
γ αγ n β is loxodromic for all small n. Moreover if q is a fixed point of β, we
have q 6= p∗ , but limn→+∞ γ −n αγ n β(q) = p∗ .
Suppose instead that q is a fixed point of α, and of γ −n αγ n β for all small
n. Again q 6= p∗ and then β(q) 6= p∗ . As above, q cannot be a fixed point of
γ −n αγ n β for all small n.
Lemma 2.1.4.
points u and v.
Suppose α is a loxodromic transformation with fixed
(i) Given p∗ 6= u, v and T > 2, there exists ε > 0 such that if β is any
loxodromic transformation with fixed points p, q satisfying d(p, p∗ ) < ε,
d(q, p∗ ) < ε, and with trace satisfying |tr β| ≥ T , then α and β generate
a classical Schottky group.
(ii) Given p, q 6= u, v, there exists T > 2 such that if β is any loxodromic
transformation with fixed points p, q and satisfying |tr β| ≥ T , and if α
also satisfies |tr α| ≥ T , then α and β generate a classical Schottky group.
640
DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
Proof. A loxodromic transformation β acts in H3 ∪ ∂ H3 . If P ⊂ H3 is
a plane orthogonal to its axis, so is β(P ). The two circles ∂P and ∂β(P ) in
∂ H3 that separate the fixed points p and q of β bound what we will refer to
as an annular region A for β. Given any point q ∗ 6= p, q, u, v in ∂ H3 , there are
annular regions for β that contain q ∗ in their interior.
Fix p∗ ⊂ ∂ H3 distinct from q ∗ , p, q, u, v. Let (pn , qn ) be a sequence with
pn 6= qn and lim pn = lim qn = p∗ . Let Tn be the transformation with fixed
point q ∗ such that Tn (p) = pn , Tn (q) = qn . Ultimately Tn is loxodromic, its
attractive fixed point converges to p∗ , and |tr Tn | → ∞. Consider an annular
domain A for β containing q ∗ in its interior. Tn A is an annular region for
Tn βTn−1 , all containing q ∗ . The sequence of bounding circles of Tn A converge
to the point p∗ ; that is, Tn A converges to ∂ H3 \ {p∗ }. The analysis would be
equally applicable to a family of transformations {β}, each with fixed points
p, q, so long as they all satisfied |tr β| ≥ T for some T > 2 (uniformly loxodromic).
Now let A0 be an annular domain for α containing p∗ in its interior. Ultimately the bounding circles of Tn A also lie in the interior of A0 . For such
indices n, α and Tn βTn−1 generate a classical Schottky group. Part (i) follows
at once.
To establish part (ii), note that both α and β have annular domains whose
boundaries are circles arbitrarily close to their fixed points, if T is large enough.
p∗ ,
Corollary 2.1.5.
Suppose γ is loxodromic with attractive fixed point
repulsive fixed point p∗ , and α, β are loxodromic as well.
(i) If α(p∗ ) 6= p∗ and β(p∗ ) 6= p∗ there exists N ≥ 0 such that γ −n αγ n and
β generate a classical Schottky group for all n ≥ N .
(ii) If α(p∗ ) 6= p∗ and β(p∗ ) 6= p∗ there exists N ≥ 0 such that γ −n αγ n and
β generate a classical Schottky group for all n ≤ −N .
Proof. This is a corollary also of Lemma 2.1.3. In case (i), the fixed points
of γ −n αγ n are arbitrarily close to p∗ for large n, since p∗ is not fixed by α, where
p∗ is not fixed by β. In case (ii), the fixed points of γ −n αγ n are arbitrarily
close to p∗ , for small n.
2.2. Lemmas regarding half-rotations.
Lemma 2.2.1. Suppose α and β each have two fixed points and β sends
one of the fixed points of α to the other. Then α likewise sends one of the fixed
points of β to the other if and only if
tr2 α = tr2 β.
641
MONODROMY GROUPS
Proof. We may assume that
µ
α=
λ
0
0
λ−1
¶
µ
and β =
¶
b
,
d
0
c
√
where λ 6= ±1 and bc = −1. The fixed points of β are (−d ± d2 − 4)/2c.
Suppose α sends one to the other. Each case implies and is implied by one of
the relations
p
d(λ2 − 1) = ± d2 − 4(λ2 + 1).
Squaring, we get d2 λ2 = (λ2 + 1)2 , or
tr β = d = ±(λ + λ−1 ) = ±tr α.
Lemma 2.2.2.
An element J of order two interchanges the fixed points
of an elliptic or loxodromic transformation γ if and only if
JγJ = γ −1 ,
and fixes them if and only if
JγJ = γ.
It fixes the fixed point of a parabolic transformation γ if and only if
JγJ = γ −1 .
Proof. For the first part we may assume that
µ
γ=
λ
0
while for the second,
µ
γ=
0
λ−1
1
0
b
1
¶
µ
and J =
¶
µ
and J =
0
−b−1
i
0
¶
b
,
0
¶
0
.
−i
The conclusion is verified by computation.
Lemma 2.2.3. Suppose α and β are loxodromic without both fixed points
in common. J is an element of order two.
(i) If J interchanges the fixed points of both α and β, J neither interchanges
nor fixes the fixed points of βα.
(ii) If J interchanges the fixed points of β but not of βαk for some k 6= 0,
then Jβ does not interchange the fixed points of α.
(iii) If J interchanges the fixed points of both βαk and βαk+1 for some k, then
J interchanges the fixed points of βαk for all k, but neither interchanges
nor fixes the fixed points of α, and does not interchange the fixed points
of αm β for m 6= 0.
642
DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
Proof. For (i), JβαJ = β −1 α−1 6= α−1 β −1 , βα.
For (ii), J1 = Jβ has order two, J1 6= id. If J1 αk J1 = α−k , then Jβαk J =
−k
α β −1 , a contradiction.
For (iii), the hypotheses Jβαk J = α−k β −1 and Jβαk+1 J = α−k β −1 JαJ
imply in turn that
αk β −1 JαJ = α−k−1 β −1 ,
or
JαJ = βα−1 β −1 ,
(6= α−1 , α).
Then
α−k β −1 = Jβαk J = JβJβα−k β −1 ,
or
JβJ = β −1 . Now, for any k,
Jβαk J = β −1 βα−k β −1 = α−k β −1 .
Finally, for any m 6= 0,
Jαm βJ = βα−m ββ −1 = βα−m β −2 6= β −1 α−m .
(Note the proof holds as well if some βαk is parabolic, under appropriate
interpretation; see Lemma 2.2.2.)
Lemma 2.2.4.
Suppose α has two fixed points but α2 6= id, while J is
an element of order two that does not interchange the fixed points of α. Then
(αJ)2 6= id and (Jα)2 6= id.
Proof. We may assume that
Ã
α=
λ
0
0
λ−1
!
Ã
and J =
b
c
−a
with λ2 6= ±1, a2 + bc = −1. Then
Ã
(αJ)2 =
!
a
λ2 a2 + bc
λ2 ab − ab
ac − λ−2 ac
bc + λ−2 a2
,
!
.
If (αJ)2 = id, then
ab(λ2 − 1)
=
0,
ac(1 − λ−2 )
=
0.
Either a = 0 or b = c = 0.The former case is impossible by hypothesis. If
b = c = 0, then since a2 = −1, we get λ2 = λ−2 = 1. This is again a
contradiction.
MONODROMY GROUPS
643
Lemma 2.2.5.
Suppose both J and J1 J interchange the fixed points of
the loxodromic or elliptic transformation γ. Then J1 fixes the fixed points of γ.
Proof. Under the hypothesis, if p, q denote the fixed points of γ, we have
J(p) = J1 J(p) and J(q) = J1 J(q). Hence J(p) = q and J(q) = p are fixed by
J1 .
Remark 2.2.6. Suppose α and β are loxodromic without a common fixed
point and β does not send one fixed point of α to the other. If γβ fixes or
interchanges the fixed points of α, then γβ −1 has neither of these properties.
In the latter case, γαβα−1 does not send one fixed point of α to the other.
What will prevent us from making use of such facts as these is that if γβ, for
example, is the θ-image of a simple loop, then in general γβ −1 and γαβα−1
are not.
A. The Pants decomposition
3. Finding a handle
3.1. Handles. By a handle H = ha, bi we mean two simple loops a, b ∈
π1 (R; O), crossing at O but otherwise disjoint, and such that α = θ(a) and
β = θ(b) are loxodromic and generate a nonelementary subgroup hα, βi of Γ.
Proposition 3.1.1. There exists a handle in R.
Proof. The proof will occupy the remainder of this chapter.
3.2. Case 1. There exists a simple, nondividing loop a ∈ π1 (R; O) for
which θ(a) = α is loxodromic. Choose b ∈ π1 (R; O) such that b is a simple
loop crossing a exactly at O, and set β = θ(b).
Suppose first that β neither interchanges the fixed points of α nor shares
a fixed point with α. Then, by Lemma 2.1.1, βαk is loxodromic for some
k. Moreover, hα, βαk i is nonelementary. We can consequently choose H =
ha, bak i.
Next suppose that β shares exactly one fixed point p with α. Because Γ
is not elementary, there is a simple loop y ∈ π1 (R; O) that does not cross a or
b and such that θ(y) = η does not fix p. Take y with the orientation such that
ay is homotopic to a simple loop. For any k, the loop ay is homotopic to a
simple loop that crosses bak exactly at O (Figure 1).
Now αη does not share the fixed point p of βαk . For at most one value of
k, αη shares another fixed point q of βαk . For if
αη(q) = q = βαk (q) = βαk+m (q)
644
DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
with m 6= 0, we have α(q) = q, and then η(q) = q = β(q), a contradiction
since q 6= p. Nor can αη send the fixed point p of βαk to another fixed point
q = αη(p) of βαk for more than one k. For
αη(p) = q = βαk (q) = βαk+m (q)
with m 6= 0 implies that α(q) = q, and then β(q) = q. This is impossible
since q 6= p. Thus there exists k such that αη neither interchanges the fixed
points of βαk , nor fixes any. By Lemma 2.1.1, we may also assume that βαk
is loxodromic.
Consequently we can return to the case above with bak and ay.
ay
y
bak
Figure 1.
Finally, suppose that β either fixes both fixed points of α or interchanges
them. Again find a simple loop y that does not cross a or b and such that
η = θ(y) neither fixes both fixed points of α nor interchanges them. Take the
orientation of y so that yb is homotopic to a simple loop. Then ηβ neither fixes
both fixed points of α nor interchanges them. Consequently we can return to
one of the cases above with a and yb.
3.3. Case 2. There is a simple, nondividing loop a ∈ π1 (R; O) such that
θ(a) = α is parabolic. Let b ∈ π1 (R; O) be a simple loop that crosses a exactly
at O.
If β = θ(b) does not fix the fixed point p of α, then βαk is loxodromic for
all large |k|, by Lemma 2.1.1. Thus we are back to Case 1.
Suppose instead that β(p) = p. There is a simple loop y ∈ π1 (R; O), not
crossing a or b, and such that η = θ(y) does not fix p. We may take y with the
orientation for which yb is homotopic to a simple loop, and hence also ybak
is homotopic to a simple loop. Since ηβ(p) 6= p, we conclude that ηβαk is
loxodromic for some k, and ybak brings us, once again, back to Case 1.
MONODROMY GROUPS
645
3.4. Case 3. Let {ai , bi } be a canonical basis for π1 (R; O), with θ(ai ) = αi
and θ(bi ) = βi . Assume that all the elements αi , βi , αj αi , βj βi , and βj αi are
elliptic or the identity. As the basis of our analysis of this case, we will find a
simple dividing loop d for which θ(d) is loxodromic.
In this section we will establish some useful lemmas.
Lemma 3.4.1. If α and β are elliptic, and their axes are not coplanar in
H3 , then βα is loxodromic.
Proof. Let P denote the plane in H3 spanned by the axis of α and the
common perpendicular l to that and the axis of β. Form the “open book”
for P with spine along the axis of α and angle half the rotation angle of α.
Then α = Rl Rlα , where Rlα and Rl are half-rotations (180◦ ) about the lines
orthogonal to the axis of α indicated in Figure 2. Similarly, β = Rlβ Rl , where
lβ is the line orthogonal to the axis of β at its intersection with l, and lies
halfway between l and β(l).
P
l
ax(a)
la
Figure 2. Open book for plane P
Consequently, βα = Rlβ Rlα . Therefore βα is elliptic if and only if the lines
lα and lβ intersect in H3 : if instead they meet at ∂ H3 , then the composition βα
is parabolic, and if they do not meet at all in H3 ∪ ∂ H3 , then the composition
is loxodromic. Since the axis of β does not lie in P , lβ does not lie in the plane
spanned by lα and l. Therefore lα and lβ cannot meet anywhere.
Corollary 3.4.2. Under the hypotheses of Case 3, the axes of the nonidentity elements of {αi , βj } either :
(a) all pass through some point ζ ∈ H3 , or
(b) all lie in a plane P ⊂ H3 , or
(c) are all orthogonal to a plane P ⊂ H3 .
646
DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
Proof. Apply Lemma 3.4.1 to the set {αi , βj }.
Note that, in case (c), the plane P contains all the lines lαi and lβi . This
is the fuchsian case: all elements of Γ preserve P .
Case (a) does not arise for our situation since Γ is nonelementary.
Lemma 3.4.3. Suppose α and β are elliptic with distinct axes that lie in
a plane P ⊂ H3 . Assume βα is also elliptic. Its axis cannot lie in P .
Proof. The axes of βα and α are different, so there is a fixed point x of
βα not lying in the axis of α. Set y = α(x); then β(y) = x. Let the plane
P 0 be the perpendicular bisector of the line segment [x, y]. By construction, x
and y are equidistant from P 0 . But x and y are also equidistant from the axis
of α, since α is a rotation about its axis. All points equidistant from x and y
lie in P 0 , so the axis of α is contained in P 0 . Since x and y are also equidistant
from the axis of β, this line, too, is contained in P 0 . We conclude that P 0 = P ,
so x ∈
/ P.
In fact, the proof shows that if the axis of βα meets P or ∂P , it does so
at, and only at, a point of intersection or common endpoint of the axes of α
and β.
Lemma 3.4.4. Suppose α, β, and γ = βα are elliptic with distinct axes,
and that they preserve a plane P ⊂ H3 . Then β −1 α−1 βα is loxodromic.
Proof. Let a, b, c denote the fixed points in P of α, β and γ. Replace α
and β by the inverses, if necessary, so that they rotate counterclockwise about
a and b. Let R1 = J, R2 and R3 denote the reflection in the lines through
[a, b], [b, c] and [c, a], respectively. Then α = R1 R3 , β = R2 R1 , and γ = R2 R3 .
The vertex angles of the triangle in Figure 3 represent the half-rotation angles.
Then
β −1 α−1 βα = JγJγ.
a
R1
b
R3
R2
c
Figure 3. Reflection triangle for α, β, β ∗ α
647
MONODROMY GROUPS
In order to better study JγJγ, we take the line l through a and b to be
the real diameter in the disk model of P (Figure 4). J is reflection in l; let
R denote reflection in the vertical line through c and Jc. Let θ denote the
half-rotation angle of γ. Let l1 denote the line through c subtending angle
θ with the vertical, and set l2 = Rl1 . Let R10 denote reflection in l1 and R20
reflection in l2 .
Jl1
J R
q
R1
l1
J
q
l
q
R2
l2
Figure 4. Reflection in Jl1 and l2
Now we have γ = R10 R = RR20 and
JγJ = JR10 JJRJ = R10∗ R,
where R10∗ = JR10 J is reflection in the line Jl1 . Consequently,
JγJγ = R10∗ RRR20 = R10∗ R20 .
The lines Jl1 and l2 cannot intersect in P ∪ ∂P . Therefore the composition of
reflections in them, R10∗ R20∗ , is loxodromic (hyperbolic).
Note, however, that R10∗ R10 = JγJγ −1 = (βα2 β)−1 can sometimes be
elliptic.
3.5. Case 3 (continued ). Suppose that the elements {αi , βi }, which are all
elliptic or the identity, preserve a plane P ⊂ H3 (Case (c) of Corollary 3.4.2).
We may assume that α1 6= id.
Consider first the case that β1 is elliptic and its fixed point in P differs
from that of α1 . Then the transformation δ = β1−1 α1−1 β1 α1 , which corresponds
−1
to the simple loop d = b−1
1 a1 b1 a1 , is hyperbolic (Lemma 3.4.4). Because d
divides R, there exists an element c of {a2 , b2 , . . . , ag , bg } with γ = θ(c) 6= id.
Apply the Dehn twist of order n about d to the simple loop cb1 , to get cdn b1 d−n .
Its image γδ n β1 δ −n is loxodromic for all large |n| by Lemma 2.1.3, since the
fixed points on ∂P of the hyperbolic δ are necessarily different from those of
the elliptics γ and β1 in P . Since cdn b1 d−n is a simple, nondividing loop, we
can return with it to Case 1 (§3.2).
648
DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
Consider next the case where β1 has the same fixed point in P as α1 , or
is the identity. We can find c in {a2 , b2 , . . . , ag , bg } such that γ = θ(c) does
not have the same fixed point in P as α1 . If θ(cb1 a1 ) is not elliptic, return
with cb1 a1 to Case 1 or 2. Otherwise, set d = (cb1 )−1 a−1
1 (cb1 )a1 , and apply
the Dehn twist about d to ca1 for a sufficiently high power. As above, return
to Case 1 with the result.
Next, suppose that the axes of the elliptic elements {αi , βi }, which are all
elliptic or the identity, lie in a plane P ⊂ H3 (Case (b) of Corollary 3.4.2). We
may assume that α1 6= id.
Assume first that the axes of α1 and β1 differ. If they cross at p ∈ P ,
or meet at p ∈ ∂P , there is an element c of {a2 , b2 , . . . , ag , bg } such that the
axis of γ = θ(c) does not contain p. By Lemma 3.4.3, the axis of β1 α1 does
not lie in P , but it crosses P at p or meets ∂P at p. Consequently this axis is
not coplanar with the axis of γ, which lies in P . Now Lemma 3.4.1 says that
γβ1 α1 is loxodromic. Return to Case 1 with cb1 a1 .
On the other hand, suppose that the axis l1 of α1 and the axis l2 of β1
are disjoint in P ∪ ∂P . If the axis l of β1 α1 is not coplanar with l1 or l2 , the
situation is again as above. If l is coplanar with each of l1 and l2 , it cannot meet
P ∪ ∂P . The plane P 0 orthogonal to l and to P is necessarily orthogonal to l1
and l2 . If the axes of all nonidentity elements of {α1 , β1 , . . . , } are orthogonal
to P 0 , we can return to the first subcase of this section. Otherwise the axis
of some γ ∈ {α2 , β2 , . . .} is not orthogonal to P 0 . Then γβ1 α1 is loxodromic,
since the axis of γ is contained in P .
Finally we need to consider the situation where β1 = id or the axes of α1
and β1 coincide. Find δ in {α2 , β2 , . . .} distinct from the identity and having
an axis distinct from that of α1 . Replace β1 by δ in the analysis above. The
triple of loops in π1 (R; O) giving rise to the loxodromic element found there
also corresponds to a simple nondividing loop, and it is only this property that
is needed.
In light of Corollary 3.4.2, the analysis of Case 3 is complete. A handle
exists, and Proposition 3.1.1 is proved.
4. Cutting the handles
4.1. We have found a special handle H as specified in §3. The next step is
to cut all the other (topological) handles, ending up with a (connected) surface
of genus one with 2(g − 1) boundary components. In cutting the handles, we
will require that the θ-image of each cutting loop is loxodromic.
Although H, or rather the established properties of the θ-image of its
fundamental group, serves to anchor the cutting process, in fact H itself will
have to undergo successive changes. It will become more and more complicated in terms of an initial basis of π1 (R; O). Roughly speaking, we will be
649
MONODROMY GROUPS
applying Dehn twists of possibly high order to felicitous combinations of simple
loops. The process will be governed by the applicability of the lemmas of §2
to yield loxodromic transformations, yet still arising under θ from simple loops
in π1 (R; O).
4.2. Let H = ha, bi denote the special handle found in Chapter 3, and
set α = θ(a), β = θ(b). We claim that after replacing H = ha, bi by another
handle of the form habq , bi or hb, abq i, if necessary, we can assume that β does
not send one fixed point of α to the other.
For suppose β sends one fixed point of α to the other. Then, using
Lemma 2.1.1(ii), find q so that αβ q is loxodromic and tr2 αβ q 6= tr2 β. Necessarily, αβ q does not share either of its fixed points with β. By Lemma 2.2.1, at
least one of the following statements is true: β does not send one fixed point
of αβ q to the other, or αβ q does not send one fixed point of β to the other.
4.3. Now suppose that hx, yi is another pair of loops in π1 (R; O) of the
form x = u−1 x0 u, y = u−1 y 0 u, where x0 and y 0 are simple loops disjoint from a
and b, with one intersection point where they cross, and u is a simple arc from
a ∩ b = O to x0 ∩ y 0 , otherwise disjoint from a, b, x0 , y 0 (see Figure 5).
b
y’
u
a
x’
Figure 5. Connection to handle H
Consider d = ybak and its θ-image δ = ηβαk , for some k. Set ξ = θ(x)
and η = θ(y). The effect of a Dehn twist of order n about d is
hα, βαk i
7→
hδ n α, βαk i,
hξ, ηi
7→
hδ n ξ, ηi.
We claim that there exist k and n such that:
(i) βαk is loxodromic;
(ii) δ = ηβαk is loxodromic;
(iii) δ n α is loxodromic without a common fixed point with βαk ;
(iv) δ n ξ is loxodromic;
650
DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
or that, after necessary relabeling and rearrangement to be spelled out below,
analogous properties hold. This claim will be established in the four steps of
§4.4.
Once this is accomplished, we will replace the handle H = ha, bi by the
handle hdn a, bak i, and cut R along dn x, represented by a freely homotopic simple loop. This operation will also serve as the basis of an induction procedure.
Note that it may well be that η = id, or ξ = id, or both. In the former case,
property (ii) is satisfied with (i), and in the second, property (iv) is satisfied
with (ii).
4.4. Step (i). By §4.2 and Lemma 2.1.1(i), there exists K ≥ 0 such that
βαk is loxodromic for all |k| ≥ K.
Step (ii). If ηβ does not interchange the fixed points of α, then by
Lemma 2.1.1 we may take K in step (i) so large that δ = ηβαk is loxodromic
for all k ≥ K or for all k ≤ −K.
If, however, ηβ does interchange the fixed points of α but ξβ does not,
interchange η and ξ and return to the paragraph above.
Finally, if both ηβ = J and ξβ = J1 interchange the fixed points of α, then
−1
ηξ = JJ1 fixes them (and is either loxodromic or the identity). In this case
replace hx, yi by hx, yx−1 i, and η by ηξ −1 , and revert to the original notation.
For this case, then, ηβαk is loxodromic for all |k| ≥ K, for some K.
Step (iii). First note that for no k ∈ Z and no m 6= 0 can both δ n α and
δ n+m α have fixed points in common with βαk . For the relations
δ n α(p)
δ n+m α(p)
= p = βαk (p),
= p = δ n δ m α(p)
imply that α(p) is a fixed point of δ, then that p is a fixed point of α, and
finally that p is a fixed point of β. The last consequence is impossible.
For any sequence k → +∞, according to Lemma 2.1.2 the fixed points of
δ = ηβαk converge to ηβ(q) and p, where q and p denote the attractive and
repulsive fixed points of α, respectively. Thus, if α sends one fixed point of δ to
the other for this sequence, then ηβ(q) = p. Similarly, for a sequence k → −∞,
the fixed points of δ converge to ηβ(p) and q. If, for this sequence, α sends one
fixed point of δ to the other, then ηβ(p) = q. By our construction, ηβ does
not interchange the fixed points of α, so α cannot interchange the fixed points
of δ = ηβαk for both a sequence k → +∞ and a sequence k → −∞.
Now if ηβ(p) = q, so that δ is loxodromic for all k ≥ K (step (ii)), then for
sufficiently large K, α cannot send one fixed point of δ = ηβαk to the other.
Likewise, if ηβ(q) = p so that δ is loxodromic for k ≤ −K, again α cannot send
MONODROMY GROUPS
651
one fixed point of δ to the other, for sufficiently large K. If ηβ sends neither
fixed point of α to the other, α sends neither fixed point of δ to the other, for
all large |k|.
We conclude that there exists K ≥ 0 such that δ = ηβαk is loxodromic
for any k ≥ K or any k ≤ −K, or both. Furthermore, α does not send one
fixed point of δ to the other. Given k in the admissible range, there exists
N = N (k) ≥ 0 such that δ n α, for all |n| ≥ N , is loxodromic and does not have
a fixed point in common with βαk .
Step (iv). Consider ξ and δ = ηβαk for fixed k ≥ K or k ≤ −K, according to (iii).
If ξ does not interchange the fixed points of δ, we can take N so large that
either δ n ξ or δ n ξ −1 is loxodromic for n ≥ N = N (k).
Suppose instead that ξ interchanges the fixed points of δ but not of
ηβαk+1 = δ 0 . Then replace δ by δ 0 .
However, not both ξ and ηξ (nor equivalently, ξ and η −1 ξ) can interchange
the fixed points of both ηβαk and ηβαk+1 . For, if so, we apply Lemma 2.2.5
to J = ξ, J1 = η (or η −1 ) and to both ηβαk and ηβαk+1 . That implies that
the fixed points of both ηβαk and ηβαk+1 coincide with fixed points p, q of η.
For this to occur, α fixes both p and q, and then ηβ must do so as well. But
since η itself fixes them, β must also fix them. This is impossible.
We may assume one of yx or y −1 x is a simple loop. Depending on which,
replace hx, yi by hyx, yi or hy −1 x, yi. Correspondingly, replace ξ by ηξ or η −1 ξ.
This returns us to one of the previous cases for δ = ηβαk or ηβαk+1 .
4.5. Cutting the surface. The loop dn x is freely homotopic to a simple,
nondividing loop d0 , disjoint from dn a and bak . Cutting R along d0 results in a
new surface R1 with a handle H = hdn a, bak i and two boundary components
freely homotopic to dn and yx−1 d−n y −1 (or y −1 x−1 d−n y). The corresponding
transformations are δ n ξ and ηξ −1 δ −n η −1 (or η −1 ξ −1 δ −n η), which have the
same trace. The common trace, however, can be made as large as desired
(Lemma 2.1.1).
If the genus of R1 exceeds one, repeat the process using the new H, and
so on. At the end, we will have a surface Rg−1 with a handle H = ha, bi (using
again the original notation) and 2(g − 1) boundary components.
Orient all the boundary components so that Rg−1 lies to their right. Let
x, y, . . . denote simple loops from the basepoint O parallel to them but otherwise disjoint from each other and from a and b. Our construction allows us
to assume that the θ-images θ(x), θ(y), . . . are all loxodromic. Pairwise they
have the same trace, but the traces of different pairs can be assumed to be
different.
652
DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
5. The pants decomposition
5.1. We carry on from the situation left in §4.5. To start, adjust the
special handle H = ha, bi as in §4.2 so that β = θ(b) does not send one fixed
point of α = θ(a) to the other. Orient b so that it crosses a from the right side
of a to the left; then the boundary of Rg−1 lies to the left of c = b−1 a−1 ba and
we have oriented the boundary so that c lies to its right.
Choose simple loops x, y ∈ π1 (Rg−1 ; O), each parallel to a boundary component and disjoint from each other and a, b, except at O = a ∩ b. The
orientations are such that ybak and xbak (but not yb−1 ak or xb−1 ak for k 6= 1)
are homotopic to simple loops for all k (see Figure 6). From §4.5 we know that
ξ = θ(x) and η = θ(y) are loxodromic.
y
b
O
x
ak
Figure 6. Connection of boundary to handle H
5.2. We begin by sorting out the following possibilities.
(1) If exactly one of ηβ and ξβ interchanges the fixed points of α, assume
that the one that does is ξβ. In this case, we claim that, for all sufficiently
large |k|, the composition δ = ηβαk does not fix either fixed point p, q of ξ.
For if ηβαk fixes p for two values of k, then p itself must be fixed by α,
and then by ηβ as well as by ξ. On the other hand, since ξβ interchanges
the fixed points p and p0 of α, we get ξβ(p0 ) = p = ξ(p), so β(p0 ) = p. This
contradiction to the known properties of the handle H establishes the claim.
(2) If neither ηβ nor ξβ interchanges the fixed points of α, then by interchanging ξ and η and relabeling if necessary, we may assume that for all
sufficiently large |k|, the composition δ = ηβαk does not fix both fixed points
p, q of ξ.
For suppose ηβαk fixes p, q for two values of k, and, correspondingly, ξβαk
fixes the two fixed points of η for two other values of k. The first supposition
implies that p and q are fixed by α, then by ηβ, and of course by ξ. The second
implies that p and q are fixed in addition by ξβ and η. But ηβ(p) = p = η(p)
implies that β itself fixes p, a contradiction.
MONODROMY GROUPS
653
It is important to note that if, in addition, ηβ sends one fixed point of α
to the other, then we may assume that δ = ηβαk , for |k| large, does not fix
even one fixed point of ξ. This is another application of the reasoning of (1).
(3) We defer consideration until §5.5 of the remaining case that both ηβ
and ξβ interchange the fixed points of α.
5.3. In this section and the next we will work with cases (1) and (2) of
§5.2. That is, ηβ does not interchange the fixed points of α, and, for all large
|k|, the composition δ = ηβαk does not fix both fixed points of ξ, and if ηβ
sends one fixed point of α to the other, ηβαk does not fix either fixed point
of ξ . Consider d = ybak , for some k, and its θ-image δ. The effect of a Dehn
twist of order n about d is
hα, βαk i
7→
hδ n α, βαk i,
hξ, ηi
7→
hδ −n ξδ n , ηi.
We will find k and n such that:
(i) βαk is loxodromic;
(ii) δ = ηβαk is loxodromic;
(iii) δ n α is loxodromic and has no common fixed point with βαk ;
(iv) δ −n ξδ n η is loxodromic and has no common fixed point with η;
(v) δ −n ξδ n and η generate a classical Schottky group.
(vi) |trδ −n ξδ n η| is unbounded in |n|.
Once this is accomplished, we will replace the handle ha, bi with hdn a, bak i,
then remove from Rg−1 the pants determined by
(d−n xdn , y, d−n xdn y),
and repeat the process.
5.4. Step (i). The properties of the special handle H (§5.1) and Lemma
2.1.1(i) imply that there is K ≥ 0 such that βαk is loxodromic for all |k| ≥ K.
Step (ii). Since ηβ does not interchange the fixed points of α, we can
choose K above so large that ηβαk is loxodromic for k ≥ K, k ≤ −K, or both.
In addition, for the admissible range of k, the composition ηβαk does not fix
both fixed points of ξ.
Step (iii). This is identical with step (iii) of §4.3. There exists K ≥ 0
such that δ = ηβαk is loxodromic for any k ≥ K, or any k ≤ −K, or both.
654
DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
Given k in the admissible range, there exists N = N (k) ≥ 0 such that, for
all |n| ≥ N , the element δ n α is loxodromic and has no fixed point in common
with βαk .
Step (iv). Note that we may assume that K is sufficiently large so that
δ = ηβαk has no fixed point in common with η for |k| ≥ K. For, if ηβαk fixes
a fixed point p of η for two values of k, then α itself must fix p, and then β
must as well.
Case (1). ηβ sends one fixed point of α to the other. In this case δ has no fixed
point in common with ξ (§5.3). Thus, by Lemma 2.1.3, the composition
δ −n ξδ n η is loxodromic for all large |n|, while we must have |n| ≥ N to
ensure that δ n α is loxodromic.
Case (2). ηβ does not send one fixed point of α to the other. Then δ n α is
loxodromic for all |n| ≥ N , while δ −n ξδ n η is loxodromic either for n ≥ N
or for n ≤ −N , for sufficiently large N .
Finally, δ −n ξδ n η and η have a fixed point in common only if δ −n ξδ n and
η do. The fixed points of δ −n ξδ n are δ −n (p) and δ −n (q), where p and q are
the fixed points of ξ. If neither point is fixed by δ, then, for sufficiently large
|n|, neither δ −n (p) nor δ −n (q) will be fixed by η. On the other hand, if p, say,
is fixed by δ, the same conclusion holds because δ and η do not share a fixed
point.
Step (v). Since δ and η have no fixed point in common, it follows from
Lemma 2.1.3 and Corollary 2.1.5 that δ −n ξδ n and η generate a classical Schottky group for sufficiently large N . Also the trace of δ −n ξδ n η can be made
arbitrarily large, for sufficiently large N .
5.5. Now we turn to the case, left aside in §5.2, where both ηβ = J and
ξβ = J1 interchange the fixed points of α. Then η = JJ1 ξ, where JJ1 is
loxodromic or the identity, and fixes the fixed points of α.
At the start we arranged matters so that ybak is homotopic to a simple
loop for all k. This is equally true of (bak )−1 y(bak ), and of x(bak )−1 y(bak ),
which is homotopic to a simple loop bounding a triply connected region (pants)
with boundary components corresponding to x and y.
We claim that, in the present case, there exists K ≥ 0 such that the
corresponding transformation
δ = ξ(βαk )−1 η(βαk ) = ξα−k β −1 Jαk
is loxodromic for all |k| ≥ K.
For ξ has no fixed point in common with α: Indeed, ξ(p) = p = α(p)
would imply that J1 β −1 (p) = p, in other words that β(q) = p, where q is the
MONODROMY GROUPS
655
other fixed point of α. Similarly, β −1 J has no fixed point in common with α.
Hence the assertion follows from Lemma 2.1.3.
We can take K so large that, in addition, ξ does not have a fixed point p
in common with
(βαk )−1 η(βαk ) = α−k β −1 Jαk
for |k| ≥ K. For, since neither ξ nor β −1 J has a fixed point in common with
α, we have on the complement of q∗ , q ∗ ,
lim α−k β −1 Jαk = q∗
k→+∞
and
lim α−k β −1 Jαk = q ∗ ,
k→−∞
where q∗ 6= p and q ∗ 6= p denote the repulsive and attractive fixed points of α.
For sufficiently large K as dictated by Corollary 2.1.5, ξ and (βαk )−1 η(βαk )
generate a classical Schottky group for all |k| ≥ K.
As in §5.4(v), the trace magnitude of the transformation (βαk )−1 η(βαk )ξ
corresponding to the new boundary component of the pants can be made arbitrarily large, in particular in comparison with that of ξ and η, which correspond
to the boundary components on which the new pants was built.
Replace the handle ha, bak i by its conjugate h(bak )−1 a(bak ), bak i. The
new pants is determined by hx, (bak )−1 y(bak )i.
5.6. The pants decomposition. In §§5.3–5.4 we showed that, given any
two boundary components of Rg−1 , we could construct a pants with them
as boundary components and such that the transformation corresponding to
the third boundary component has trace of arbitrarily large magnitude. The
surface remaining after this pants is removed is again of genus one, but with one
fewer boundary components. Again choosing any two boundary components,
we can construct another pants, and so on until all that remains is a surface
of genus one with one boundary component: a handle.
For later requirements, we will specify the initial steps of the decomposition as follows: Group the 2(g − 1) boundary components of Rg−1 into pairs,
where the two components of each pair arise from cutting a handle of R. Construct first (g − 1) pants, one corresponding to each pair, which then comprise
two of its boundary components. After this is done, finish the construction
with any possible succession of pairings.
Each pair of boundaries of Rg−1 corresponds to transformations of the
same trace, but we may assume from §4.5 that different pairs correspond to
transformations of different traces. When each new boundary component forming a new pants is inserted, we can ensure by §5.4(v) and §5.5 that the trace
magnitude of its corresponding transformation exceeds that corresponding to
all previously inserted boundaries.
The combinatorics of the decomposition and a corresponding reorganization of the generating set for π1 (R; O) will be discussed in §5.8 below.
656
DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
5.7. The final cut. We are left with 2g−3 pants and a handle H. Yet more
adjustment to H is necessary before gaining the assurance that one final cut
will produce a pants decomposition {Ri } called for in §1.7. Consider the handle
H = ha, bi remaining at the end of the process. A simple loop c ∼ b−1 a−1 ba
bounds H on its right side (starting in §5.1 we specifically assumed that b
crosses a from the right side of a to the left). On the left side of c is a pants
with boundary components oriented so that the pants is to their left. Also,
c−1 ∼ yx, where x and y are simple loops parallel to the boundary components,
and are disjoint from c, b and a except for a shared basepoint.
Set α = θ(a), β = θ(b), σ = θ(c), ξ = θ(x), η = θ(y). We know that hξ, ηi
is a Schottky group and that α and β are loxodromic without a common fixed
point. As in §4.2, we can assume that β does not send one fixed point of α to
the other.
By construction (see §5.4(iv)–(v) and §5.5), the trace magnitude of ηξ
exceeds that of η and ξ. In particular, neither ξ nor η can be conjugate within
PSL(2, C) to ηξ = α−1 β −1 αβ = σ −1 .
We may assume that ξβ = J does not interchange the fixed points of α,
and is not the identity. Otherwise, replace ξ and η by their conjugates σ m ξσ −m
and σ m ησ −m , where m is chosen so that σ m ξσ −m β = Jm neither interchanges
the fixed points of α nor is the identity. To see that such an m exists, consider
σ m ξσ −m ξ −1 = Jm J, which either has the same fixed points as α, or has order
two. The latter case is impossible because hξ, ηi is a Schottky group. Since ξ
and σ = ηξ have no fixed points in common, the former is impossible as well,
except perhaps for a finite number of values of m. Consequently, replace hξ, ηi
by the conjugate group hσ m ξσ −m , σ m ησ −m i, and correspondingly hx, yi by the
conjugate pants hcm xc−m , cm yc−m i. Return again to the original notation.
Now we are ready to cut the handle H. But first, apply a Dehn twist of
order k about a. This changes H to ha, bak i.
Next, apply a Dehn twist of order n about a simple loop d ∼ xbak . This
results in the changes
ha, bak i
7→
hdn a, bak i.
Finally, cut the resulting handle along a simple loop freely homotopic to
bak . This results in a pants whose fundamental group is
hbak , (dn a)−1 (bak )−1 (dn a)i.
We claim that k and n can be chosen so that the groups representing the
adjacent pants are now both Schottky groups
hγ, (δ n α)−1 γ −1 (δ n α)i
where γ = βαk and δ = ξγ.
and hξ, δ −n ηδ n i,
MONODROMY GROUPS
657
(i) There exists K ≥ 0 such that δ = ξβαk is loxodromic either for k ≥ K
or for k ≤ −K; for definiteness assume the former is true. Indeed we have
already arranged matters so that ξβ does not interchange the fixed points
of α.
(ii) For sufficiently large K and k ≥ K the composition δ = ξγ has no
fixed points in common with ξ or γ, and αγα−1 has no fixed points in common
with αβα−1 β −1 ξ −1 6= id.
First note that neither βαk nor αβαk−1 can have the same fixed points
for two values of k. For example, βαk (p) = p = βαm (p) for m 6= k implies that
α(p) = p, and then β(p) = p, which is impossible. Consequently, for sufficiently
large K and k ≥ K, the element γ = αβ k does not share a fixed point with
ξ or η, nor αγα−1 with αβα−1 β −1 ξ −1 , provided this latter is not the identity.
It follows that neither ξγ and γ, nor ξγ and ξ, can share fixed points either.
Finally, αβα−1 β −1 ξ −1 6= id because ξ is not conjugate to ηξ = α−1 β −1 αβ
(because they have unequal traces, as we have seen earlier in §5.7).
(iii) We show that hξ, δ −n ηδ n i is a Schottky group, either for all n ≥ N
or all n ≤ −N , for some N ≥ 0; for definiteness we will assume the former.
For if δ has both its fixed points in common with η, then δ and η commute
and the group remains hξ, ηi. If δ has one fixed point in common with η, say its
repulsive fixed point p, the fixed points of δ −n ηδ n converge to p as n → +∞.
Since p is not also a fixed point of ξ, the group is Schottky for large n. If δ
has no fixed points in common with η, it is Schottky for all large |n|.
(iv) We show that hγ, (δ n α)−1 γ −1 (δ n α)i is a Schottky group for all
|n| ≥ N , for sufficiently large N in (iii) and fixed k ≥ K from (i) and (ii).
For the fixed points of (δ n α)−1 γ −1 (δ n α) are the images under α−1 δ −n
of the fixed points of γ. As n → +∞ or n → −∞, these images converge to
α−1 (p), where p is the repulsive or attractive fixed point of δ, since δ and γ have
no fixed points in common. If α−1 (p) is not a fixed point of γ, Corollary 2.1.5
implies that the group is Schottky for large |n|.
Suppose to the contrary that γα−1 (p) = α−1 (p), while ξγ(p) = p. Then
αγα−1 (p) = p, while p = αγα−1 γ −1 ξ −1 (p) = αβα−1 β −1 ξ −1 (p). This does not
occur, by (ii).
Remark 5.7.1. Had we not been so concerned about the final cut forming
two of the boundary components of a single pants corresponding to a Schottky
group, we would have proceeded more simply, as follows. Cut H = ha, bi along
a resulting in a pants ha, b−1 a−1 bi. Pair boundary components of this with
those of neighboring pants hx, yi, to get two new pants ha, yi and hb−1 a−1 b, xi.
Apply to these the Dehn twist of order m about c ∼ b−1 a−1 ba. For all large
|m|, the corresponding groups are easily seen to be Schottky.
658
DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
5.8. The combinatorics of pants decomposition. We will systematically
organize a generating set for the fundamental group of R in terms of the pants
decomposition {Pi }.
Start by fixing points Oi , Oi0 , Oi00 on each component of ∂Pi , and disjoint
simple auxiliary arcs from Oi to Oi0 and Oi00 . In terms of these auxiliary arcs,
there is a unique path in Pi between any two boundary components. Also,
a component of ∂Pi with an assigned orientation uniquely determines a loop
from Oi , which we will take as the basepoint of π1 (Pi ; Oi ). If ai and bi are
two boundary components of Pi , an orientation of ai uniquely determines an
orientation of bi such that bi ai is homotopic to a simple loop around the third
(here making use of the auxiliary arcs).
If the components a of ∂Pi and a0 of ∂Pj correspond to the same simple
loop on R, choose the points O ∈ a and O0 ∈ a0 to correspond to the same
point on R.
Let T denote the trivalent graph of genus g corresponding to the pants
decomposition {Pi }: each vertex of T corresponds to one of the pants Pi , and
each edge corresponds to a pair (a, a0 ) of boundary components, one on each
pants corresponding to an endpoint. Two boundary components are paired
(a, a0 ) if and only if they correspond to the same simple loop on R.
T has 2g − 2 vertices and 3g − 3 edges. Exactly g of the vertices have
one-edge loops attached to them; this is a consequence of the particular combinatorics of the decomposition. We call these vertices extreme.
Remove from T those g one-edge loops; the result T0 is a maximal (connected) tree. The extreme vertices of T are those that are extreme in T0 in
the sense that only one edge of T0 is attached to the vertex.
Designate one of these extreme vertices as the root v0 of T0 : for example,
the vertex corresponding to the last handle we cut. There is a unique simple
path in T0 from any vertex to the root.
Denote the pants corresponding to the vertex v by P (v). Consider the
vertices v 0 6= v whose shortest path to v0 contains v. Mark the boundary
components of P (v) where these shortest paths first cross; we will use these
shortest paths below. If v is not extreme, two of the three boundary components of P (v) will be marked. If v is extreme but v 6= v0 , none of the boundary
components will be marked. Exactly one of the boundary components of P (v0 )
will be marked.
Making use of the auxiliary arcs in the {Pi }, the simple edge-arc in T0
from the vertex vi = Pi to vj = Pj uniquely determines a simple arc in R from
Oi to Oj . Likewise, a simple edge-loop in T uniquely determines a simple loop
in R.
Let P0 be the pants corresponding to the root v0 , and O = O0 the designated basepoint for its fundamental group. Take also O as the basepoint of the
fundamental group of R. As we have seen, T0 uniquely determines a simple arc
MONODROMY GROUPS
659
ci in R from O to each Oi . Thus, a simple loop ai ∈ π1 (Pi ; Oi ) can be uniquely
associated with c−1
i ai ci ∈ π1 (R; O). Suppose ei is one of the g edge-loops of
T, with both end points on the same vertex vi . Likewise with the help of the
auxiliary arcs in Pi = P (vi ), the edge ei , with an assigned orientation, uniquely
0
determines a loop c0−1
i ei ci ∈ π1 (R; O).
The totality of elements c−1
i ai ci from oriented boundary components of
0
/ T0 generate π1 (R; O).
pants {Pi } plus g elements c0−1
i ei ci from edges e ∈
B. Pants configurations from Schottky groups
6. Joining overlapping plane regions
6.1. In this section we will describe a method of using covering surfaces
to separate two overlapping plane regions which are acted on by a common
Möbius transformation. It is no restriction to describe the process with the
loxodromic transformation α : z 7→ λ2 z, with |λ| > 1 and fixed points 0 and
∞. Let T or T (α) denote the quotient torus
T = (C \ {0})/hαi,
and π the projection from C \ {0}. Denote the simple compressing loop π({z :
|z| = 1}) in T by c. A noncontractible simple loop on T lifts to a closed curve
in C \ {0} if and only if it is freely homotopic (or homologous) to ±c.
If a simple loop a is not of this type, a∗ = π −1 (a) is a simple α-invariant
arc. If a is given the orientation dictated by α, the arc a∗ is directed toward
the attractive fixed point.
Conversely, if a∗1 is a simple, α-invariant arc in C directed toward the
attractive fixed point, a1 = π(a∗1 ) is a simple loop freely homotopic (or homologous) to the result of applying to a the Dehn twist about c of some order n:
namely, a1 ∼ a + nc.
6.2. Let SN denote the N -sheeted cover of the sphere S1 = S2 , branched
over the fixed points 0 and ∞ of α. Topologically, SN is again a sphere. The
map z 7→ z 1/N = w sends S N back to S1 ; it is conformal except at 0 and
∞. The cyclic group of cover transformations is conjugated to the group of
rotations hw 7→ e2πi/N wi.
The transformation α lifts to an automorphism α∗ of SN , determined up
to composition with cover transformations. It is conjugated to the loxodromic
transformation w 7→ λ2/N w, which in turn is determined only up to composition with cover transformations.
Consider the torus TN = TN (α), defined by
TN = (SN \ {0, ∞})/hα∗ i.
660
DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
It is the N -sheeted torus over T , uniquely determined by the properties that
a lifts to exactly N mutually disjoint simple loops and cN lifts to one simple
loop.
For the following lemma a and c are simple loops on T as before: c is the
projection of the unit circle and a is the projection of a simple, α-invariant arc
a∗ ⊂ C, positively oriented by α. If the simple arc a1 crosses a transversely at
every point of intersection, the geometric intersection number is defined as the
number of points of intersection. We assume this number is finite.
Lemma 6.2.1. Suppose a1 is freely homotopic and transverse to a, with
geometric intersection number n. Set N = 2n + 1. Then there is a lift a0 of
a and a lift a01 of a1 that are disjoint, freely homotopic simple loops in TN .
Correspondingly, there is a lift a∗ of a and a∗1 of a1 to SN that are disjoint,
α∗ -invariant simple arcs.
Remark 6.2.2. A more precise measure of intersection would be to set
n = max |m(τ )|,
where τ ⊂ a1 is a segment whose endpoints don’t lie on a, m(τ ) is the algebraic intersection number of τ and a, and the maximum is taken over all such
connected segments τ of a1 .
Proof. Fix a lift a0 of a to TN or a lift a∗ to SN . We can label the
N = 2n + 1 sheets on TN or on SN over T \ {a} in cyclic order, starting to
the left of a0 or of a∗ . A point in the (n + 1)-st sheet can be connected to
one on a0 or on a∗ only by crossing n other lifts of a. Fix p ∈ a1 \ a, and the
point p0 or p∗ lying over p in the (n + 1)-st sheet. The endpoint of the arc ã1
lying one-to-one over a1 \ {p} and starting at p0 or p∗ also lies in the (n + 1)-st
sheet, because a1 is freely homotopic to a; in TN , the arc ã1 closes up to form
a simple loop. The conclusion is a direct consequence.
Note that without the condition that a1 , positively oriented by α, be freely
homotopic to a, the conclusion of the lemma is false. Instead, the following is
true.
Corollary 6.2.3. Suppose, more generally, that the simple loop a1 ⊂ T ,
transverse to a, is the projection of an α-invariant arc in C \ {0}. There exists
N = N (a, a1 ) ≥ 1 and m ∈ Z such that δ m a1 and a have disjoint lifts on TN
and SN , where δ denotes the Dehn twist about c.
6.3.
Lemma 6.3.1. Suppose a and a1 are α-invariant simple arcs in C \ {0},
the lifts of freely homotopic transverse loops in T (α) with geometric intersection number n. Suppose a is contained in the boundary of a simply connected
MONODROMY GROUPS
661
region P ⊂ C \ {0} lying to its left, while a1 is contained in the boundary of
a simply connected region P1 lying to its right. Set N = 2n + 4. Then on
SN \ {0, ∞} there are disjoint lifts a∗ of a and a∗1 of a1 with the property that
the corresponding lifts P ∗ of P and P1∗ of P1 that contain a∗ and a∗1 in their
respective boundaries are disjoint as well.
Proof. Fix a lift a∗ of a in SN \ {0, ∞} and let E be the generator of the
order-N cyclic group of cover transformations with the property that a∗ and
Ea∗ bound to the left of a∗ a lift σ ∗ of C \ {a}, which we will refer to as the
first sheet of the covering. In cyclic order to the left of a∗ the lifts of a are
Ea∗ , . . . , E N −1 a∗ , and the corresponding sheets are σ ∗ , Eσ ∗ , . . . , E N −1 σ ∗ .
Denote by P ∗ the lift of P adjacent to a∗ on its left side. Necessarily P ∗
lies entirely in the first sheet σ ∗ .
If a1 is disjoint from a, then N = 4 (although N = 3 will do). Let a∗1 be
the lift of a1 lying in the third sheet E 2 σ ∗ , and P1∗ the lift of P1 adjacent to
a∗1 on its right side. P1∗ lies in the sector bounded by a∗1 and E −1 a∗1 , which lies
in the second sheet Eσ ∗ . Hence P1∗ is disjoint from P ∗ (see Figure 7).
Ea*
P*
s*
a*
P1*
a1*
E3a*
Figure 7. Separation of regions when N = 4
More generally, choose p ∈ a1 ∩ a and let p∗ denote the point over p on
E n+2 a∗ . Let a∗1 denote the lift of a∗1 through p∗ ; then a∗1 does not intersect
E 2 a∗ or E −2 a∗ . Consequently, E −1 a∗1 does not intersect Ea∗ . Let P1∗ be the
lift of P1 adjacent to a∗1 on its right side; P1∗ lies in the sector between a∗1 and
E −1 a∗ . Therefore P1∗ is disjoint from P ∗ .
Note that we have not optimized the choice of N , which can be done in
particular cases.
Corollary 6.3.2. In the hypotheses of Lemma 6.3.1, assume that not
only a and a1 but also P and P1 are α-invariant in C \ {0}. There is a lift α∗
of α to SN that leaves P ∗ and P1∗ invariant.
662
DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
Proof. Let α∗ be the lift of α that maps the first sheet σ ∗ onto itself, and
hence P ∗ onto itself. Necessarily α∗ maps every sheet E k σ ∗ onto itself, and
hence P1∗ onto itself as well.
6.4. Joining overlapping regions. In this section we will build a prototype
for the procedure that forms the basis of §8. It is typical of tricks used in classical function theory and is a generalization of a technique applied to Möbius
groups called grafting [Mas1], [He1], [Go1].
Consider the hypotheses of Lemma 6.3.1: a and a1 are α-invariant simple
arcs in C \ {0} directed toward ∞, and one does not spiral around the other
(an informal way of saying that they arise from freely homotopic loops in T ).
The region P lies to the left of a, and P1 to the right of a1 . Like a and a1
themselves, P and P1 can badly overlap each other.
However, on SN , P ∗ and P1∗ are disjoint. Let Q∗ be the region on SN that
lies to the right of a∗ and to the left of a∗1 : then P1∗ ∪ a∗1 ∪ Q∗ ∪ a∗ ∪ P ∗ is a
simply connected region R∗ in SN \ {0, ∞}. According to Corollary 6.3.2, if P
and P1 are α-invariant, α∗ is a conformal automorphism of R∗ .
Let g : H2 → R∗ be a Riemann map, where the hyperbolic plane H2 is
realized as the unit disk. Then g −1 α∗ g is a hyperbolic Möbius transformation
α0∗ in H2 . Let π : SN → S2 denote the projection. Then f = π ◦ g is a locally
univalent meromorphic function on H2 with the property that
f α0∗ (z) = αf (z)
for all z ∈ H2 . That is, f determines a complex projective structure on H2
that induces the isomorphism between cyclic groups hα0∗ i → hαi.
We have joined together the annular regions P/hαi = P ∗ /hα∗ i and P1 /hαi
∗
= P1 /hα∗ i by means of the annulus Q∗ /hα∗ i, which attaches to the boundary
components a/hαi and a1 /hαi.
7. Pants within rank-two Schottky groups
7.1. Suppose hα1 , α2 i is a two-generator classical Schottky group acting
on its regular set Ω ⊂ S2 . The quotient surface R = Ω/hα1 , α2 i has genus two
(and bounds the handlebody R+ = H3 /hα1 , α2 i if the group is extended to
hyperbolic three-space).
There are round circles b∗1 and b∗2 , mutually disjoint in Ω, with the following
property: The two pairs of circles (b∗1 , α1 b∗1 ) and (b∗2 , α2 b∗2 ) are mutually disjoint
with mutually disjoint interiors in S2 , and αi maps the exterior of b∗i onto the
interior of αi b∗i . The circles b∗1 and b∗2 are lifts of mutually disjoint, nondividing
simple loops b1 and b2 in R. These bound disks in R+ and for that reason are
called compressing loops.
663
MONODROMY GROUPS
Let a1 and a2 be simple, nondividing loops in R such that a1 ∩(a2 ∪b2 ) = ∅
and a2 ∩ (a1 ∪ b1 ) = ∅, while ai crosses bi transversely at a single point. Then
a1 and a2 have lifts a∗1 and a∗2 to Ω uniquely determined by the condition that
they are α1 -invariant and α2 -invariant simple arcs, respectively. Let δi denote
the Dehn twist about bi . Then, for example, δ1n a1 can be used in place of a1 : it,
too, can be taken to be a simple loop disjoint from a2 and b2 , meeting and there
crossing b1 at a single point. It too has a uniquely determined α1 -invariant lift
(δ n a1 )∗ in Ω. (More generally, the simple loop a01 has an α1 -invariant lift if
and only if a01 is freely homotopic to a1 within the handlebody R+ .)
7.2. Finding pants. Assign a1 and a2 their positive orientation, that is,
the one that directs a∗1 and a∗2 , their α1 - and α2 -invariant lifts, toward the
attractive fixed points of α1 and α2 . We can join a1 and a2 to a common
basepoint O ∈ R so that the resulting simple loops a01 and a02 have the property
that a02 a01 is homotopic to a simple loop a03 ; this loop a03 is then freely homotopic
to a simple loop a3 that, together with a1 and a2 , divides R into two pants P
and P 0 ; also, a3 has an α2 α1 -invariant lift a∗3 and an α1 α2 -invariant lift α1 a∗3
(Figure 8).
Note that the free homotopy class of a3 on R is not uniquely determined
by that of a1 and a2 : we can change a3 by applying Dehn twists about a simple
0−1 0 0
dividing loop homotopic to b0−1
1 a1 b1 a1 without affecting a1 or a2 . We can
also change a3 by applying Dehn twists about b1 or b2 , but that will change
a1 or a2 as well. In any case there is an α2 α1 -invariant lift of a3 .
b2
b1
a2
a1
P’
P
a3
b*2
b*1
a1*
a 1 a*3
a*2
a 1 b*1
a3*
P0*
a 2 b2*
Figure 8. Pants determined by Schottky group
664
DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
Let P denote the pants lying to the right of a1 and a2 , and to the left of
a3 . Of course we may assume that b1 ∩ P and b2 ∩ P are simple arcs. There
is a lift P0∗ of P to Ω that is an “octagon” bounded by connected segments of
a∗1 , a∗2 , a∗3 , α1 a∗3 and b∗1 , b∗2 (see Figure 8, bottom). The orbit of P0∗ (adding its
boundary arcs on b∗1 and b∗2 ) under hα1 , α2 i is a simply connected region P ∗
that is the universal cover of P .
7.3. Isomorphisms are geometric. We summarize the analysis of §7.2 as
follows.
Lemma 7.3.1.
Let Q be a pants with oriented boundary components
(d1 , d2 , d3 ), and choose generators d01 , d02 , d03 , d03 ∼ d02 d01 , for π1 (Q; O) such
that d0i is parallel to di , O ∈ Q. Suppose θ is the isomorphism of π1 (Q; O)
onto the Schottky group hα1 , α2 i determined by the correspondence θ(d01 ) = α1 ,
θ(d02 ) = α2 . Then there is a pants P in R = Ω/hα1 , α2 i bounded by simple
loops (a1 , a2 , a3 ), positively oriented by α1 , α2 , α2 α1 , and a homeomorphism
h : Q̄ → P̄ taking di (with its orientation) to ai , i = 1, 2, 3, which induces θ:
there is a point O∗ ∈ P ∗ ⊂ Ω over h(O) ∈ P such that the lift of h(d0i ) from
O∗ terminates at αi (O∗ ), for i = 1, 2, 3.
Proof. In §7.2 we observed the following convention for finding pants P
and P 0 in a Schottky group with designated generators α1 and α2 . The three
boundary components have α1 -, α2 -, and α2 α1 -invariant lifts, positively oriented by α1 , α2 and α1 α2 , respectively. If a1 and a2 are represented by generators a01 and a02 in π1 (P ; O) or π1 (P 0 ; O0 ), then a02 a01 is homotopic to a simple
loop parallel to a3 . The two pants P and P 0 are distinguished in that one lies
to the right of a1 and a2 , and to the left of a3 , while the opposite holds for the
other.
The orientations of the di can temporarily be reversed as necessary so that
Q lies to the right of d1 , d2 and left of d3 . Make the corresponding temporary
replacements of αi by αi−1 . Now find a pants P meeting the requirements, and
then return to the original designations.
7.4. Two groups with a common generator : Compatibility. Consider two
Schottky groups hα1 , α2 i and hα2 , α3 i with a common generator α2 . Denote the
regular sets in S2 by Ω and Ω0 , and set R = Ω/hα1 , α2 i and R0 = Ω0 /hα2 , α3 i.
Choose simple loops (a1 , b1 , a2 , b2 ) in R and (a02 , b02 , a03 , b03 ) in R0 as in §7.1; here
a0j and ai are taken with their positive orientations from αj and αi . Find, as
in §7.2, a pants P ⊂ R lying, say, to the right of a2 , and then a pants P 0 ⊂ R0
lying to the left of a02 .
665
MONODROMY GROUPS
Definition 7.4.1. As above, suppose a2 and a02 are simple loops on R and
0
R0 , with α2 -invariant lifts a∗2 and a0∗
2 to Ω and Ω , respectively. The loops a2
0
and a2 are compatible (with respect to α2 ) if the projections of a∗2 and a0∗
2 (that
0
is, the embeddings of a2 and a2 ) in the torus T (α2 ), are freely homotopic there.
Recall that T (α2 ) = (S2 \ {p, q})/hα2 i, where p and q are the fixed points
of α2 . Let δ2 denote the Dehn twist about b02 on R0 . In general a02 will not be
compatible with a2 . However,
Lemma 7.4.2. The loop a02 on R0 can be made compatible with a2 on R:
a2 is compatible with δ2m a02 for a unique value of m.
Proof. Let δ2 denote the Dehn twist about b02 on R0 . Note that b02 embeds
as a simple loop on T (α2 ), so that δ2 can be taken to act on T (α2 ) as well as
on R0 . For exactly one value of m, the loop δ2m a02 will be compatible with a2 .
Remark 7.4.3. We emphasize that the process of making a2 and a02 compatible affects only one of the surfaces: say R0 . And, on R0 , it affects only a02 ,
not a03 . However, the third boundary component c0 of the pants P 0 is affected.
Indeed, there is a lift c0∗ to Ω0 invariant under α3 α2 . The simple loop b02 which
crosses c0 once also embeds in T (α3 α2 ), and the twist δ2 equally can be taken
to act on the torus T (α3 α2 ). Thus, under the action of δ2m on R0 , the loop c0
changes to δ2m c0 ; the pants δ2m P 0 is bounded by δ2m a02 , δ2m c0 , and a03 .
7.5. Compatibility conditions on one pants. Consider a Schottky group
hα, βi and a pants P in Ω/hα, βi, as in Figure 9.
Denote the boundary components of P by a, b, c, with the orientation
indicated. With respect to these curves, we can find compressing loops x and
y (which lift to closed loops in Ω), with the orientations and intersections
indicated in the figure.
c
x
a
y
b
Figure 9. Pants and compressing loops
666
DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
A Dehn twist of order p about x composed with a twist of order q about
y has the following effect on a, b, c:
a 7→ δ p+q a,
b 7→ δ −q b,
c 7→ δ p c.
Here we use the notation δ k t to denote the effect on the simple oriented loop t
of a twist of order k about an oriented simple compressing loop crossing t once,
from its right side to its left; geometrically the result is realized and accounted
for on the torus T (τ ) that is associated with t.
Suppose b and c are to be paired with boundary components b0 and c0
on other pants, where δ m b is compatible with b0 and δ n c is compatible with
c0 . This can be fulfilled simultaneously in P by setting p = n and q = −m.
The effect on a is to replace it by δ n−m a. That is, compatibility for two
boundary components of P can always be achieved, but then the state of the
third boundary component is determined.
Suppose instead that c is to be paired with c0 on another pants, with
compatibility requirement c0 = δ n c, while b is to be paired with a with compatibility requirement a = δ m b. In terms of §7.4, this means that there is a
transformation γ with α = γβγ −1 , where a and b have been determined by α
and β, respectively. That is, there is an α-invariant lift a∗ and a β-invariant
lift b∗ , and the two can be compared in terms of the α-invariant arcs a∗ and
γb∗ .
Therefore p = n, while q is determined by the condition
−q = n + q + m,
or q = − 12 (m + n).
A solution q ∈ Z exists if and only if m + n is even, that is, if m and n have
the same parity.
In other words, the algebraic sum [(p + q) − q + p] = 2p of the Dehn twists
that can be applied effectively to the boundary components of a pants is even.
Consequently, if the requirements for compatibility in a pants demand that the
algebraic sum be odd, those requirements cannot be met.
Remark 7.5.1. There is also a compressing loop u in Ω/hα, βi that divides,
separating a and b while crossing c twice. A Dehn twist about u leaves a and b
unchanged, but changes the homotopy type of c and P on the surface Ω/hα, βi.
Yet it leaves unchanged the free homotopy type of the projection c∗ of c to its
associated torus T (βα).
For on T (βα), there are two representatives of u, u∗1 and u∗2 . They are
disjoint, parallel and oriented opposite one another: one crosses c∗ from right
to left, the other from left to right. A Dehn twist about u is reflected by twists
on T (βα) about u∗1 and u∗2 . But, because of their opposite orientations, these
twists cancel, leaving the free homotopy class of c∗ unchanged.
In short, twists about u have no effect on compatibility questions.
MONODROMY GROUPS
667
7.6. A compatibility condition on identical pants. For later application
in §8, consider the following augmentation to the second situation of §7.5,
where α = γβγ −1 . In the conjugate group γhα, βiγ −1 , consider the pants
P1 that corresponds precisely to P , distinguishing corresponding elements by
the subscript. Suppose, as before, that c and c1 are to be paired with c0 and
c01 on other pants P 0 , P10 , but now with the same compatibility requirements:
c0 = δ n c and c01 = δ n c1 . Instead of pairing b with a as before, pair b1 with a and
b with a1 . Because the two groups are virtually identical, the compatibility
requirements are a = δ m b1 and a1 = δ m b.
The result of Dehn twists of order p and q about x and y, and of order p1
and q1 about x1 and y1 , is as calculated in §7.5. We must have p = p1 = n.
That leaves, for q and q1 , the equation
−q1 = n + q + m,
or q + q1 = −(m + n).
In this case there are always solutions: for example, q = −m and q1 = −n.
8. Building the pants configuration
8.1. What remains to be done? In §5.7 the combinatorics of the pants
decomposition {Pi } of R found in Part A was described as a trivalent graph
T arising from a tree T0 ⊂ T by the addition of g edges, one attached to each
extreme vertex. The universal cover of T is reflected in the combinatorics of
their lifts {Q∗i } in the universal cover H2 of R, that is, in how the lifts fit
together.
Corresponding to each lift Q∗i is the Schottky group θ(Stab Q∗i ), which
in turn stabilizes the lift Pi∗ of a pants Pi in its quotient surface. Using the
technique of §6, our goal is to follow the information in T, or the combinatorics
of {Q∗i } in H2 , to build a simply connected Riemann surface J. This will be
the universal cover of a surface obtained by joining together the pants {Pi } by
attaching auxiliary cylinders.
However, to join a boundary component a of Pi to a0 of Pj (or perhaps
to a0 of Pi ), it is necessary that a and a0 be compatible in the sense of §7.4.
It is not necessarily true that the totality of compatibility conditions can be
satisfied.
In §§8.2–8.4, typical cases of joining pants will be described, before we
draw the general conclusions in §§8.4–8.6. In §9, we will show how to add
branch points when needed.
8.2. Joining pants. We continue with the situation of §7.4. There, we
found simple loops a3 ∼ a2 a1 on R and a04 ∼ a03 a02 on R0 such that (a1 , a2 , a3 )
bound pants P ⊂ R lying to the right of a1 and a2 , while (a02 , a03 , a04 ) bound
668
DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
pants P 0 ⊂ R0 lying to the left of a02 and a03 . Here (a1 , a2 , a02 , a03 ) are positively
oriented by generators α1 , α2 and α3 . According to Lemma 7.4.2, the loop a02
can be taken compatible with a2 . We will now show how to join the pants P
and P 0 by attaching a cylinder to the left side of a2 and the right side of a02 .
Let P ∗ denote the region in Ω over P and P 0∗ the region over P 0 in
Ω0 . Both P ∗ and P 0∗ are simply connected, as they represent the respective
universal covers.
We are in a position to apply Lemma 6.3.1 to P ∗ and P 0∗ . There exists an
N -sheeted SN of S2 , branched over the fixed points of α2 , on which there are
0∗∗
∗
0∗
∗∗ and P 0∗∗ of
disjoint lifts a∗∗
2 and a2 of a2 and a2 that border disjoint lifts P
P ∗ and P 0∗ : the projections P ∗∗ → P ∗ and P 0∗∗ → P 0∗ are homeomorphisms.
Equally well, P ∗∗ and P 0∗∗ represent the universal covers of P and P 0 .
Next, take the sector Q∗∗ on SN lying between the left side of a∗∗
2 and the
,
and
form
right side of a0∗∗
2
∗∗
∗∗
0∗∗
0∗∗
Q∗∗
∪ a∗∗
.
1 =P
2 ∪ Q ∪ a2 ∪ P
∗
Then Q∗∗
1 is invariant under a lift α2 of α2 to SN . It comes with a conformal
∗
2
structure and a projection π into S which is a locally injective meromorphic
function.
∗∗ generated by the cover
Construct the orbit of Q∗∗
1 under the group Γ
transformations of P ∗∗ over P and P 0∗∗ over P 0 ; Γ∗∗ is the free product of these
groups with amalgamation over hα2∗ i. This can be done as follows. Suppose,
/ hα2∗ i is a cover transformation of P ∗∗ over P , so that
for example, that α∗ ∈
α∗ is the lift of a cover transformation α of P ∗ over P . In particular, α∗ sends
∗∗ to the edge α∗ a∗∗ , which is invariant under the conjugate
the edge a∗∗
2 of P
2
α∗ α2∗ α∗−1 of α2∗ .
∗∗ at a∗∗ . We correspondingly
But the configuration Q∗∗
1 extends beyond P
2
∗ a∗∗ . Moreover there is a projection π ∗ of
)
to
extend
beyond
α
attach α∗ (Q∗∗
1
2
2
Q∗∗
1 into S which is a local homeomorphism, the extension of the restriction
of π ∗ to P ∗∗ . Extend π ∗ from P ∗∗ to α∗ (Q∗∗
1 ) by
π ∗ (z) = απ ∗ (z0 ),
z = α∗ (z0 ),
z0 ∈ Q∗∗
1 .
The cover transformation γ ∗ of P ∗∗ or P 0∗∗ over P or P 0 is conjugated to
the cover transformation α∗ γ ∗ α∗−1 of α∗ (P ∗∗ ) = P ∗∗ or α∗ (P 0∗∗ ) over P or
P 0 . The transformation α∗ γ ∗ α∗−1 itself is the lift of the cover transformation
αγα−1 of P ∗ over P or of α(P 0∗ ) over P 0 .
∗∗
∗
∗∗
Q∗∗
1 and then Q1 ∪ α (Q1 ) are simply connected Riemann surfaces that
inherit their complex structure from S2 via π ∗ .
Continuing on, we construct a pants configuration
J(P, a2 ; a02 , P 0 )
which is a simply connected Riemann surface with a group of conformal automorphisms Γ∗∗ . It has a meromorphic projection π ∗ into (usually onto) S2 ,
MONODROMY GROUPS
669
which is a local homeomorphism. The projection π ∗ induces a homomorphism
of Γ∗∗ onto the group generated by hα1 , α2 i and hα2 , α3 i.
Consequently, with the group Γ∗∗ , the abstract configuration
J(P, a2 ; a02 , P 0 )
is a model for the universal covering of the Riemann surface
P ∪ a2 ∪ (Q∗∗ /hα2∗ i) ∪ a02 ∪ P 0 .
It is a four-holed sphere; the pants P and P 0 have been connected by the
cylinder Q∗∗ /hα2∗ i, which joins a2 and a02 .
The Riemann mapping
g : H2 → J(P, a2 ; a02 , P 0 )
conjugates Γ∗∗ to a fuchsian group G in H2 . The function f = π ∗ g : H2 → S2
is meromorphic and locally univalent in H2 . It gives a projective structure on
the four-holed sphere H2 /G with the associated homomorphism sending G to
the group generated by hα1 , α2 i and hα2 , α3 i.
8.3. Adding to the join of two pants. At the level of the pants P in R and
P 0 in R0 , the construction of §8.2 only involved neighborhoods of the boundary
components a2 of P and a02 of P 0 , and the sector of SN between their two lifts.
Thus, suppose there is another Schottky group hα3 , α4 i sharing the generator α3 with hα2 , α3 i. We can join the boundary component a03 of P 0 to a
compatibly chosen boundary component a003 of a pants P 00 in R00 = Ω00 /hα3 , α4 i,
lying to the right of a003 , by constructing the appropriate SN . A lift of P 0∗ appears in both configurations J(P 0 , a03 ; a003 ; P 00 ) and J(P, a2 ; a02 , P 0 ), and these two
lifts of P 0∗ can be identified.
Join together these two configurations by identifying the two lifts P 0∗∗ and
∗∗
P10∗∗ of P 0∗ . After that, further construct its orbit under Γ∗∗
2 . Now Γ2 is the
∗∗
∗∗
0
0
00
free product of Γ and the corresponding group Γ1 of J(P , a3 ; a3 , P 00 ) with
amalgamation over the common subgroup Stab(P 0∗∗ ) = Stab(P10∗∗ ), which is
just the lift of the covering group of P 0∗ over P 0 . We end up with an abstract
configuration
J(P, a2 ; a02 , P 0 , a03 ; a003 , P 00 ) = J2 ,
which is a simply connected Riemann surface with a group Γ∗∗
2 of conformal
∗
2
automorphisms. There is a meromorphic projection π into S that is a local
homeomorphism and induces a homomorphism of Γ∗∗
2 onto the group generated
by hα1 , α2 i, hα2 , α3 i and hα3 , α4 i. Also, J2 is a model of the universal cover
for a five-holed sphere formed by connecting P to P 0 as in §7.5, and the result
to P 00 with an appropriate cylinder connecting a03 and a003 .
670
DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
8.4. Making handles. In §8.2, suppose that instead of a second Schottky group, we are presented with a transformation β such that βα1 β −1 = α2 .
We can as well join the group hα1 , α2 i to its conjugate βhα1 , α2 iβ −1 =
hα2 , βα2 β −1 i, to have the effect that the boundary component a2 of the pants
P in R is joined to a1 . We must start by ensuring that a∗2 is compatible with
βa∗1 with respect to α2 ; this may require replacing a2 by the result of applying
some power of a Dehn twist about b2 .
As before, we can find an SN that holds disjoint lifts of P ∗ and βP ∗ . Then
a configuration J is constructed with a group of automorphisms Γ∗∗ isomorphic
to the HNN extension of Stab(P ∗ ) by a suitable lift β ∗ of β. This J is a simply
connected Riemann surface with a locally univalent meromorphic projection
into S2 . It is a model for the universal covering surface for the one-holed
torus obtained by attaching the cylinder obtained from SN to the boundary
components a1 and a2 of P .
8.5. Recall from §5.7 the trivalent graph T and the maximal tree T0 ⊂ T.
There, we chose one of the extreme vertices of T0 as the root. Let Tr denote
the graph resulting from T after removing the one-edge loop hanging from the
root. Thus Tr represents a surface S ⊂ R of genus g − 1 with two boundary
components. Let Σr denote the subgroup of π1 (R; O) that is the fundamental
group of S.
Lemma 8.5.1. There exists a pants configuration J(Tr ) modeled on Tr .
It is a simply connected Riemann surface, the universal cover of a Riemann
surface S of genus g − 1 with two boundary components. Let g : H2 → J(Tr ) be
a Riemann mapping, and π : J(Tr ) → S2 the meromorphic projection. Then
f = πg is a projective structure for S for the homomorphism θ : Σr → θ(Σr ) ⊂
Γ.
Proof. First we check that the compatibility conditions can be satisfied.
Denote by P (v) the pants corresponding to the vertex v, and by Γ(v) the
Schottky group with regular set Ω(v).
In §5.8 we marked the boundary components of P (v) according to the
following rule. There is a unique path in Tr from any vertex v 0 to the root v0 .
The unmarked boundary component a of P (v) is the one on the path from v
itself. If v is not an extreme point of T0 , it has two immediate predecessors v1
and v2 , and P (v) has two marked boundary components b and c, lying on their
paths to v0 . Following the notation of §7.5, let x and y denote compressing
loops (which lift to simple loops in Ω(v)) such that x crosses c and a, and y
crosses b and a.
Now move down the tree T0 . Start at the extreme vertices v 6= v0 . Two
of the boundary components b and c of P (v) are to be paired. Make them
compatible by a twist about either x or y.
MONODROMY GROUPS
671
Continue down the tree. Do not go to a vertex before dealing with all its
predecessors. Arriving at a vertex v and P (v) with marked borders b and c,
replace them by the result of twists about x and y, so as to be compatible with
the (unmarked) borders b0 and c0 associated with the immediate predecessors
v1 and v2 . When the root v0 is reached, the one marked border of P (v0 ) is
made compatible with its immediate predecessor.
Finally, use the technique illustrated in §§8.2–8.4 to join the pants {Pi }
together with auxiliary cylinders to build a Riemann surface of genus g − 1
with two boundary components remaining from the pants P (v0 ). This is done
by building a pants configuration J(Tr ), which is its universal cover.
8.6 The final handle or the two-sheeted covering. Having constructed
J(Tr ), all attention is focused on P (v0 ), with its three boundary components
a, b, c and compressing loops x, y as in §7.5. Since P (v0 ) has been attached
to its predecessor, say by establishing the compatibility of c with its partner
c0 , no more twisting about x is possible. Can we make a compatible with b,
allowing attachment of the final handle? As we have seen in §7.5, this is possible if and only if one can do the job with an even number of twists. If so,
we can finish the construction of J(T), the pants configuration reflecting the
full trivalent graph T, which will then be a simply connected Riemann surface
with a group of conformal automorphisms making it the universal cover of a
surface of genus g.
If not, keeping in mind the alternate construction of §7.6, we will construct
instead a pants configuration J that models a two-sheeted unbranched covering
of the reference surface R.
Suppose a and b have arisen from cutting R along a curve b00 , freely homotopic to the nondividing simple loop b0 ∈ π1 (R; O). Set R0 = R \ {b00 },
−1
and find the simple loop a0 ∈ π1 (R; O) such that b0 and a0 b−1
0 a0 give rise to
π1 (P (v0 ); O). The group
N = ha20 , π1 (R0 ; O), a0 π1 (R0 ; O)a−1
0 i
is a normal subgroup of index two in π1 (R; O). It defines a two-sheeted unbranched covering R̃ of R that is a compact surface of genus 2g − 1.
The surface R̃ is explicitly constructed as follows. Label the boundary
−
0
components of R0 as b+
0 and b0 , corresponding to the two sides of b0 in R.
0
Take another copy R0 of R0 . Then R̃ is the surface obtained by identifying b+
0
0 with b− and b+ , respectively on R . The cover transformation
and b−
on
R
0
0
0
0
0
is determined by a0 .
Let T2 denote the trivalent graph built likewise by taking two copies of Tr
and attaching two new edges e1 and e2 , as follows. The endpoints of the new
edges are the two vertices corresponding to v0 (and pants P (v0 )), and they
serve to pair the boundary components a and b on one copy of P (v0 ) with b
and a, respectively, on the other.
672
DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
Correspondingly, take two copies of J(Tr ). Because of the compatibility
established in §7.6, they can be joined together following the combinatorics of
T2 and the restriction of θ to N . The resulting pants configuration J(T2 ) is
again a simply connected Riemann surface with a group of conformal automorphisms isomorphic to N , making it the universal cover of a surface of genus
2g − 1.
Because of the asymmetry in satisfying the compatibility for the two copies
of P (v0 ) (see §7.6), J(T2 ) does not have conformal automorphisms that represent the sheet interchange of R̃. If, however, J(T) can be constructed, and
then J(T2 ) constructed in addition, J(T2 ) will have that symmetry: it will
represent the universal cover of the two-sheeted cover of the Riemann surface
corresponding to J(T).
A Riemann mapping g : H2 → J(T) or g2 : H2 → J(T2 ) conjugates the
cover transformations to a fuchsian group G isomorphic to π1 (R; O) or to a
fuchsian group G2 isomorphic to the index two subgroup N . Let π denote
the projection of J(T) or J(T2 ) to S2 . The meromorphic function f = πg
or f2 = πg2 determines a projective structure that induces θ : G → Γ or
θ : G2 → θ(G2 ) ⊂ Γ.
We cannot exclude the possibility that θ(G2 ) = Γ. Although the transformation in Γ that makes the conjugation corresponding to the pairing of the
boundary components a, b of P (v0 ) is not the identity (because P (v0 ) arises
from a two-generator Schottky group), it may already belong to θ(G2 ). In any
case, if θ : π1 (R) → Γ cannot be lifted to SL(2,C), θ : N → θ(N ) can be so
lifted.
9. Attaching branched disks to pants
9.1. One can attach a disk to any surface with boundary by introducing a
single branch point. Explicitly for our situation, consider a pants P embedded
in C and a boundary component a oriented so that P lies to its right. Suppose
d is an oriented simple loop bounding a disk ∆ lying to its right. Suppose that
d crosses a at a point p, and that z1 and z2 are points separated by both a
and d, with z1 ∈ P ∩ ∆. Assume that there exists a simple arc σ between z1
and z2 that crosses both loops at p and is otherwise disjoint from them. Set
σ0 = σ ∩ P ∩ ∆.
Attach the ∆ to P as follows. Denote the opposite sides of σ0 by σ0+ and
−
σ0 . Identify the side σ0+ of ∆ \ σ0 with the side σ0− of P \ σ0 , and the side σ0−
of ∆ \ σ0 with the side σ0+ of P \ σ0 . This determines a new Riemann surface
P 0 that is conformally equivalent to a new pants. Its boundary ∂P 0 consists of
a ∪ d (here d lies “over” P ) and the remaining components of ∂P . The natural
holomorphic projection π : P 0 → P ∪ ∆ is a local homeomorphism except at
the point over z1 , where it behaves like z 7→ z 2 . See Figure 10.
673
MONODROMY GROUPS
z2
s
p
a
d
z1
P’
Z
2
P’
@
s
Figure 10. Attachment of branched disk
Note that the construction does not essentially depend on a choice for σ.
Instead we can work in the two-sheeted cover of S2 , branched over z1 and z2
The same construction can be applied to attach an (n − 1)-sheeted disk
to P , for any n ≥ 2.
9.2. Application to pants in a Schottky group. Suppose that hα, βi is a
Schottky group acting on Ω ⊂ C, and P ⊂ Ω/hα, βi is a pants with boundary
components a, b, c oriented so that P lies to the right of a and b, which have
α- and β-invariant lifts a∗ and b∗ in Ω. Let d be a compressing curve on the
handlebody surface Ω/hα, βi that crosses a exactly once, at a point p.
Introduce a simple arc σ in Ω/hα, βi that joins a point z1 ∈ P to z2 in its
complement, and crosses the loops a and d at p, otherwise being disjoint from
them. Set σ0 = σ ∩ P .
Let d∗ be a simple loop in Ω lying over d, which crosses a∗ (necessarily
once). Orient d and thus d∗ so that the disk ∆ lying to its right contains the
lift of σ0 that is adjacent to d∗ .
Attach ∆ to P by means of the slit σ0 . Neither the resulting pants P1
is embedded in C nor ∆ is embedded in Ω/hα, βi. Nevertheless, any annular
neighborhood of d in Ω/hα, βi is conformally equivalent to its lift about d∗ .
Thus the conformal structure of P1 is well defined.
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DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
Equivalently, the universal cover P̃ of P is embedded in Ω, and the universal cover P̃1 of P1 arises from that by attaching ∆ by means of the lift of σ0
that is adjacent to d∗ , and then taking the orbit under hαi of the attachment.
We need to examine this construction more closely.
The attachment of ∆ to a∗ at p∗ ∈ a∗ over p leads to the attachment of
the loop d∗ to a∗ at p∗ : as we move along a∗ toward the attracting fixed point
of α, when we reach p∗ we take a detour along d∗ in its positive direction,
returning to p∗ and then continuing along a∗ . Since d∗ intersects a∗ only at
p∗ , the resulting arc is essentially a simple arc, and so is its hαi-orbit, which
covers the point set a∗ ∪ αk (d∗ ).
The essentially simple, α-invariant arc a∗ ∪αk (d∗ ) can equally be described
as follows. It is the lift of the result of applying to a on Ω/hα, βi, or its
representation in the torus T (α), a Dehn twist about d.
9.3. Another alternative to the geometric obstruction of Section 8.6. In
§8.6 we faced the question of adding the final handle to the pants configuration
J(Tr ). If that was not possible, we showed that we could instead construct a
pants configuration corresponding to a two-sheeted, unbranched cover of the
surface of genus g.
Alternatively, using the construction of §9.2, we can carry out the final
construction after introducing a branch point of order two (or any even order).
That is, we can construct a pants configuration Jb (T) representing the universal
covering of a Riemann surface of genus g. If g : H2 → Jb (T) is a Riemann map,
and π : Jb (T) → S2 is the natural projection, then f = π ◦ g is a meromorphic
function. It is locally injective except at the conjugacy class of branch points
of order two, and still induces the homomorphism θ : π1 (R; O) → Γ.
10. The obstructions
10.1. The modulo 2 construction invariant. An admissible pants decomposition {Pi } for the homomorphism θ : π1 (R; O) → Γ is one for which the
restriction of θ sends each π1 (Pi ) to a Schottky group. Its combinatorics are
associated with a trivalent graph T. To each vertex v of T is associated a
Schottky group S(v) = hαv , βv i acting on Ω(v) ⊂ S2 . To each S(v) is associated a pants P (v) ⊂ Ω(v)/S(v) with boundary components a, b, c that have
αv -, βv - and βv αv -invariant lifts in Ω(v). In terms of corresponding elements of
π1 (P (v)), we have c0 ∼ b0 a0 in Ω(v)/S(v). The orientation of P (v) with respect
to a and b, and hence c, has been dictated by that of the corresponding Pi
with respect to its boundary components and carried over to T by θ.
Each edge e of T corresponds to a common generator α of the two Schottky
groups S(v1 ) and S(v2 ) if the endpoints of e lie on v2 6= v1 . If v2 = v1 ,
MONODROMY GROUPS
675
then e is associated with a pair of boundary components of P (v1 ), which in
turn correspond to generators αv and βv related by βv = γv αv−1 γv−1 for some
element γv ∈ Γ. In any case the pair of boundary components corresponding
to α project to a pair of simple loops on the torus T (α). The two boundary
components are called compatible if their projections, appropriately oriented,
are freely homotopic on T (α).
We will call T compatible if all pairs of boundary components of the associated pants {P (v)} are compatible.
Recall that on each torus T (α) there is a free homotopy class of simple
loops called compressing loops (§6.1), each of which lifts to a simple loop in S2 .
Lemma 10.1.1. Suppose that on each T (α) one of the boundary projections is freely homotopic to the result of a Dehn twist
of order´n(α) (about a
³P
compressing loop) applied to the other. Set n(T) =
α n(α) mod 2. There
is a compatible pants decomposition {P (v)} corresponding to T if and only if
n(T) = 0.
Proof. To each pair of pants one can apply Dehn twists about compressing
loops on Ω(v)/S(v). The algebraic sum n(P (v)) of their effect on the three
boundary components of P (v) is an even number. Thus
X
n(P (v)) = 0 (mod 2).
Hence the values of n(T) cannot be changed by repositioning the pants P (v)
in the surfaces Ω(v)/S(v).
For the graph T of §8.4 that represents the “localization” of the obstruction
to the construction, the question of compatibility rested on the compatibility
of the two paired boundary components in the root pants P (v0 ) (§8.6). This
was precisely the question of whether or not n(T) = 0. That is, if n(T) = 0 we
can distribute the twists so that T is compatible.
For other graphs T, we refer to Corollary 10.5.1.
10.2. Lifting Schottky groups. Lifting refers to the property that a given
homomorphism θ : π1 (R; O) → PSL(2, C) lifts to a homomorphism θ∗ : π1 (R; O)
→ SL(2, C). The image groups are not necessarily isomorphic.
It is helpful to recall the case where H = hα, βi is a two-generator, purely
loxodromic fuchsian group. As such it represents either a handle or a pants.
Let A and B be matrix representations of α and β. Then H is isomorphic
to hA, Bi. The commutator matrix [A, B] is independent of the choice of lift
of α and β. The two cases, handle or pants, can be distinguished according
to whether [α, β] represents a simple loop or not, or whether no axis in its
conjugacy class separates the axes of α and β or does, or whether tr[A, B] < −2
or tr[A, B] > 2. Moreover, in the case of a handle, the free homotopy class
676
DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
in the torus T ([α, β]) determined by a loop parallel to the handle boundary
is uniquely determined, independent of Dehn twists about compressing loops
when regarding hα, βi as a Schottky group.
More generally, any Schottky group hα, βi can be lifted to an isomorphic
group in SL(2, C) by designating matrix representatives for α and β.
10.3. The modulo 2 lifting obstruction. Let T be a trivalent graph as in
§10.1. Lift to SL(2, C) the Schottky groups corresponding to its vertices.
Let e be an edge of T with endpoints v1 and v2 . If v1 6= v2 , the edge e
corresponds to a common generator α of S(v1 ) and S(v2 ). The lifting will be
called compatible on e if the lifted α in S(v1 ) and lifted α in S(v2 ) have the
same trace. If v1 = v2 , the compatibility condition is that the designated lifts
of α and γαγ −1 from S(v1 ) have the same trace. The lifting of T will be called
compatible if it is compatible on each edge.
Suppose T is the graph of §8.4 with its maximal tree T0 . Start at the
extreme vertices of T0 and work down towards the root: Exactly in analogy to
the construction of §8.4, choose at each step a lift of a generator of a Schottky
group to be compatible with the lifts previously chosen. We end up with a
compatible lift of Tr . The lift of Tr is determined by the two choices made at
the g − 1 extreme points of T0 other than the root, and one choice at the root.
Lemma 10.3.1. Suppose T is the trivalent graph corresponding to an
admissible pants decomposition. Then T has a compatible lift to SL(2, C) if
and only if the homomorphism θ can be lifted to SL(2, C).
Proof. The graph
T corresponds to a presentation of π1 (R).
10.4. Localization of the lifting obstruction. Denote by hα−1 βα, β −1 i the
Schottky group corresponding to the root. We recall from §5.7 that the “handle
group” H = hα, βi is nonelementary with α and β loxodromic, even though it
may not be discrete.
Applying the technique of §8, we can build a pants configuration Jh on
which H acts so that Jh /H is a handle. Likewise the graph T0h resulting from
removing from T the root and attached edges determines a pants configuration
J0h acted on by a group H 0 so that J0h /H 0 is a surface of genus g − 1 with one
boundary component.
Choose matrix representatives A and B for α and β; then [B, A] is a
representative for [β, α], which corresponds to the boundary component of the
handle.
The graph T0h can be lifted to SL(2, C) as in §10.4, which yields a matrix
C representing [β, α] ∈ H 0 . Therefore C = ±[B, A].
MONODROMY GROUPS
677
Lemma 10.4.1. The homomorphism θ lifts to SL(2, C) if and only if
C = [B, A]. In particular, θ lifts if Jh and J0h can be joined to form a pants
configuration for T.
Proof. The first assertion follows from Lemma 10.3.1. The second follows
as a consequence of the existence of a projective structure (see, for example,
Lemma 1.3.1).
10.5. Equivalence of obstructions.
Proposition 10.5.1. The procedure of §8 succeeds in constructing a
projective structure associated with the given homomorphism θ : π1 (R; O) →
PSL(2, C) if and only if θ can be lifted to a homomorphism into SL(2, C).
Proof. From §1.3 we already know lifting is a necessary condition. Now
suppose θ can be lifted, yet the construction cannot be completed. That is, in
the notation of §10.4, Jh cannot be attached to J0h . But then, as in §9, we can
introduce a single branch point of order two and construct instead a branched
projective structure associated with θ. According to §1.4, θ cannot be then
lifted to SL(2, C), in contradiction with the assumption.
Corollary 10.5.2. If the construction of a projective structure works
for one admissible pants decomposition for θ, it works for any admissible decomposition.
C. Ramifications
11. Holomorphic bundles over Riemann surfaces, the 2nd
Stiefel-Whitney class, and branched complex projective structures
The purpose of this chapter is to place Theorem 1.1.1 in a more general
setting, and to use that to clarify the role played by branched structures in
Part B. We will also discuss relations between instability of holomorphic vector
bundles over Riemann surfaces and branched complex projective structures.
In §11.5 we establish the local character of the map between singly branched
structures over Teichmüller space and the representation variety. In §11.6,
we again use holomorphic vector bundles to prove that for singly branched
structures too the monodromy representation is necessarily nonelementary.
11.1. The 2nd Stiefel-Whitney class of sphere bundles over Riemann surfaces. Suppose that η : P → R is a holomorphic CP1 -bundle over a closed
Riemann surface R. It is known (see for instance [Beau, Prop. III.7]) that
P can be obtained as the projectivization of a holomorphic (rank 2) vector
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DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
bundle ξ : V → R. Let det(V ) denote the determinant bundle of V , this is
a holomorphic line bundle over the surface R. The bundle V is not uniquely
determined by the projective bundle P → R, and to obtain an isomorphic projective bundle, we can alter V by multiplying it by a holomorphic line bundle Λ
over R. Then deg(det(V ⊗ Λ)) = deg(det(V )) + 2deg(Λ). Thus we can always
choose V so that det(V ) has degree 0 or 1.
Let p : V → P (V ) = P be the projectivization. We shall think of p as
a holomorphic line bundle over the base P . It is well-known that there are
exactly two topologically distinct orientable S2 -bundles over the surface R (see
[Mel]) and they are distinguished by the 2nd Stiefel-Whitney class w2 (P ) of
the bundle P → R.
Note that if deg(det(V )) = 0 then the determinant bundle det(V ) is topologically trivial. In this case the bundle V is associated to an SL(2, C)-bundle
over R which is henceforth topologically trivial. We conclude that w2 (P ) equals
deg(det(V )) (mod 2).
Let σ : R → P (V ) be a holomorphic section of P (V ). It defines a holomorphic line bundle L → R by pull-back σ ∗ (p) of the line bundle p. The line
bundle L is canonically embedded as a holomorphic subbundle of the bundle
ξ : V → R with the image p−1 (σ(R)).
Lemma 11.1.1. (1) σ 2 = deg(det(V )) − 2deg(L), where the left-hand
side is the self -intersection number of the cycle σ(R) in P (V ). (2) The number
σ 2 (mod 2) equals the 2nd Stiefel -Whitney class w2 (P ) of the bundle η : P → R.
Proof. The first assertion is a particular case of a general result proven in
[La, §1]. Since w2 (P ) equals deg(det(V )) (mod 2), the second assertion follows.
Nevertheless we will provide a elementary proof of the first assertion for
the sake of completeness. We first consider the case deg(det(V )) = 0 and
then we shall reduce the general case to this one. If deg(det(V )) = 0 then
both bundles V and P are topologically trivial. Hence there is an orientation
preserving diffeomorphism P (V ) → R × F , where F = S2 . By the Künneth
formula, the homology class [σ] can be written as
[σ] = n[F ] + [R],
and we get: σ 2 = 2n. There are two possible cases: n ≥ 0 (if σ 2 ≥ 0) and n < 0
(if σ 2 < 0). We consider the former; the later case is analogous (one just has
to work with anti-holomorphic functions instead of the holomorphic ones). We
can think of σ : R → R × F as a graph of a smooth function f : R → F = S2
which has nonnegative degree n. The function f is not holomorphic, however
(after deforming the section σ within its homotopy class) we can assume that
f −1 (∞) = Z := {z1 , ..., zn } ⊂ R and f is holomorphic near each point zj so
MONODROMY GROUPS
679
that f 0 (zj ) 6= 0, 1 ≤ j ≤ n. Now we realize F = C ∪ {∞} as the complex
projective line CP1 so that the point ∞ has the homogeneous coordinates
[1 : 0]. Then we lift the function f to the meromorphic function
f˜ : R → C2 , f˜(z) = (f (z), 1)
which does not have zeroes and is holomorphic in a punctured neighborhood
of each point zj ∈ Z and has a simple pole at each zj ∈ Z. Thus f˜ corresponds
to a smooth meromorphic section of the line bundle L ⊂ V which has n simple
poles and no zeroes. Hence deg(L) = −n = −σ 2 /2.
Now we consider the case when deg(det(V )) = 2k is an even number. Take
a complex line bundle Λ over R so that deg(Λ) = −k, then deg(det(Λ⊗V )) = 0.
The section σ : R → P defines complex line subbundle of Λ ⊗ V which is
isomorphic to Λ⊗L. As we proved above, σ 2 = deg(det(Λ⊗V ))−2 deg(Λ⊗L)
which in turn equals to deg(det(V )) − 2 deg(L). This completes the proof in
the case when deg(det(V )) is even.
In the case when deg(det(V )) is odd take a 2-fold unramified covering
R̃ → R. Then the bundle V → R pulls back to a bundle Ṽ → R and
deg(det(Ṽ )) = 2 deg(det(V )) is even. Similarly, the section σ determines a
section σ̃ : R̃ → P (Ṽ ) and σ̃ 2 = 2σ 2 . The pull-back of the line bundle L to
L̃ ⊂ Ṽ has degree equal to 2 deg(L). We get:
σ̃ 2
=
deg(det(Ṽ )) − 2 deg(L̃)
σ2
=
deg(det(V )) − 2 deg(L).
which implies
This concludes the proof in the general case.
11.2. Branched structures. Consider a Riemann surface R = Ω/π1 (R)
where Ω is the universal cover R̃ of R and is either the unit disk, or the complex
plane, or the Riemann sphere and the group π1 (R) of Möbius transformations
acts freely and discontinuously on Ω.
Suppose that θ : π1 (R) → Γ ⊂ PSL(2, C) is a homomorphism, and
f : Ω → f (Ω) ⊆ S2 is a meromorphic function (without essential singularities)
which is θ-equivariant and defines a branched (complex ) projective structure σ
on R as in §1.4. Alternatively one can define a branched projective structure
on R as a collection of locally defined holomorphic (but not necessarily univalent) mappings φα from R to S2 so that different mappings are related by
Möbius transformations γα,β :
φα = γα,β ◦ φβ
(see for instance [Man1]).
The homomorphism θ is the (projective) monodromy representation of the
branched projective structure, and in the terminology of §1.3 the projection
680
DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
f∗ : R → S2 is the (multivalued ) developing map. We define the branching
divisor Df as follows. Consider the discrete set D̃f ⊂ R̃ consisting of critical
points of f . Thus (after holomorphic change of variables), near such a critical
point zj the function f (z) can be written as
f (z) = z k , 2 ≤ k < ∞.
The number k is the order of branch point zj . Since the function f is θequivariant we conclude that for any γ ∈ π1 (R) the point γ(zj ) is again a branch
point with the same order k. Hence the projection of D̃f to the surface R is
a finite collection of points, to each such point wj we have the associated the
number ord(wj ) = kj > 1 which is its order. Define the (additive) branching
divisor D = Df of the structure σ as
X
(kj − 1)wj .
wj
The number
d=
X
(kj − 1) = deg(Df ) ≥ 0
wj
is the degree of this divisor. The number kj − 1 is the local degree degwj (D) of
the divisor D at the point wj . The multiplicity |D| of the divisor D is just the
number of points in it. If deg(D) = 0, the divisor D is empty and there is no
branching.
For reasons that we shall see later on, it is convenient to define the divisor
D by subtracting 1 from the order of each branch point. In addition we will
consider the branching divisor as a topological object, not an analytic one.
Thus we will say that two branching divisors D, D0 on R are equivalent if there
exists a bijective order-preserving map D → D0 between them. This is the only
meaningful equivalence relation in our situation since we will have to change
the complex structure on R in order to find a branched projective structure
with the prescribed monodromy.
Next we review the relation between branched projective structures and
Schwarzian differential equations as in §1.4. Let D be a positive divisor on
the Riemann surface R. Suppose that φ(z)dz 2 is a meromorphic quadratic
differential on R which is holomorphic on R − D and near each point wj ∈ D
has a Laurent expansion of the form
(7)
φ(z) =
∞
(1 − kj2 ) b X
+
ai z i .
+
2z 2
z i=0
Here we use local coordinates such that wj = 0 and kj − 1 = degwj D is
the local degree of D. If
(8)
f (z) = z kj h(z)
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MONODROMY GROUPS
where h(z) is a holomorphic function such that h0 (0) 6= 0, then the Schwarzian
derivative Sf (z) near zero has Laurent expansion of the form (7). Conversely,
to have a solution in the form (8) the quadratic differential φ(z)dz 2 must satisfy
an extra condition of integrability; see [He1] or [Man2].
Let QD (R) denote the space of meromorphic quadratic differentials on R
with at most simple poles at points of D. Suppose that ψ0 is a fixed quadratic
differential of the form (7), then all other such quadratic differentials can be
written as φ = φ0 + ψ, where ψ ∈ QD (R). Let n denote the multiplicity of D.
There exists a collection of n polynomials Kj on the (3g − 3 + n)-dimensional
complex vector space QD so that φ is integrable if and only if the differential
ψ belongs to the zero set of all the polynomials Kj . If deg(D) ≤ 2g − 2 then
the algebraic variety
I(R, D) := {Kj (ψ) = 0, j = 1, ..., n}
has generic dimension 3g − 3. In the case of a single-order two branch point
at the orbit of z = 0 ∈ H2 , I(R, D) is given by the polynomial equation
u2 + 2bu + 2v = 0
(9)
where u is the coefficient of the z −1 term and v is the constant term in the
Laurent expansion of ψ at z = 0. The number b is given by §1.4(6). We refer
to [Man1, 2, 3] for more details.
Now we go back to the linear differential equation
u00 + 12 φu = 0
(10)
expressed in a local coordinate system on the surface R. With φ ∈ QD (R) + φ0
and satisfying the integrability condition as above, the equation (9) has two
linearly independent solutions. If zj is a singular point of φ and we choose
local coordinates so that zj = 0, near this point these solutions have the form
(
u1 (z) = z (1+kj )/2 (1 + o(1))
u2 (z) = z (1−kj )/2 (1 + o(1)).
A circuit about z = 0 generates the linear monodromy
µ
u1
u2
Ã
¶
7→ J
kj −1
u1
u2
!
,
Ã
where
J=
−1
0
0
−1
!
.
The projectivization of this monodromy in PSL(2, C) is just the identity.
Lemma 11.2.1. On the surface R−D with a base-point O, the differential
equation (9) has a linear monodromy representation
θ∗ : π1 (R − D, O) → SL(2, C).
682
DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
Proof. This is a consequence of the fact that the Wronskian of two solutions is a constant (see Corollary 1.3.1).
Let U ⊂ R be a closed disc which contains all the singular points zj ∈ D
and fix a base point O ∈ ∂U = `. The matrix θ∗ (`) that results from analytic
continuation along ` equals J d where d = deg(D) is the degree of this divisor
and J = −1. The representation θ∗ projects to a homomorphism θ : π1 (R) →
PSL(2, C). We conclude that θ can be lifted to a linear representation
θ̃ : π1 (R) → SL(2, C)
if and only if the number d is even, in particular if d = 0 as in Chapter 1. It
is instructive to see a topological proof of this fact as well.
Let P denote the S2 -bundle over R associated with the monodromy representation θ of a complex projective structure τ on R. It carries a natural
flat connection. Let w2 (θ) := w2 (P ). The developing map f of the structure
τ defines a holomorphic section σ of the holomorphic bundle P → R. We will
treat σ as a 2-cycle in P .
Proposition 11.2.2. Under the above conditions we have:
hσ(R), σ(R)i = 2 − 2g + deg(D),
where h·, ·i is the intersection pairing on the 4-manifold P .
Proof. Note that the polynomial z n admits arbitrarily small deformations
pε in the space of polynomials of degree n so that p0ε (z) has only simple roots
near zero. Thus, after perturbing the projective structure a little bit and
keeping the homomorphism θ the same, we assume that the order of each
critical point of the meromorphic function f : Ω → S2 is 2. It is clear that this
perturbation does not change hσ(R), σ(R)i and d = deg(D). The developing
section σ is transversal to the flat connection over all points of R except at the
singular points ξ1 , ..., ξd of the structure. Let D be the divisor of this singular
locus. There exists a smooth vector field X on R, which has n = 2g + 2
nondegenerate zeros, where g is the genus of R: it has 1 sink, 1 source, and 2g
saddle-type points. (For instance, take a Morse function µ : R → R which has
one minimum, one maximum and 2g saddle points, then using a Riemannian
metric on R let X := grad(µ).) Denote zeroes of X by ζ1 , ..., ζn where the last
two points have index 1. We can choose X so that
{ζ1 , . . . , ζn } ∩ {ξ1 , . . . , ξd } = ∅.
Thus the vector field X is a section of the tangent bundle TR which is transversal to the zero section. Now using the developing section σ : R → P we
lift the vector field X to a tangent vector field Y = σ∗ (X) along the surface Σ = σ(R) ⊂ P . The vertical directions in P define the normal bundle
MONODROMY GROUPS
683
N (Σ) as in subsection 11.1. The flat connection on P defines the projection
∇ : Tx (P ) → Vx (P ) where Vx (P ) is the distribution of vertical planes in P .
The vector field Q = ∇(Y ) is a section of the normal bundle N (Σ). The section σ is transversal to the flat connection on P everywhere except at the set
{ξ1 , . . . , ξd }. Thus the set of zeros of the field Q is
σ{ξ1 , . . . , ξd , ζ1 , . . . , ζn }.
A direct computation shows that the section Q of the normal bundle N (Σ)
is transversal to the zero section 0Σ . Moreover, the intersection Q(Σ) ∩ 0Σ
is positive at the points {ξ1 , . . . , ξd , ζn−1 , ζn } and is negative at the points
{ζ1 , . . . , ζn−2 }. Hence the algebraic intersection number hQ(Σ), 0Σ i (which is
equal to hΣ, Σi) equals
d + 2 − (n − 2) = d + 2 − 2g
which proves the proposition.
Corollary 11.2.3. The degree deg(D) = d is even if and only if the representation θ lifts to SL(2, C). Equivalently, θ is liftable if and only if the second
Stiefel -Whitney class satisfies the equation w2 (P ) = deg(D) = 0(mod 2).
Proof. The representation θ lifts to SL(2, C) if and only if the bundle P
is trivial (equivalently, w2 (P ) = 0); see [Go2]. As in the previous proposition
we have the developing section σ of the bundle P → R. We proved that
hσ(R), σ(R)i = 2 − 2g + deg(D); hence
hσ(R), σ(R)i = deg(D) (mod 2).
On the other hand, according to Lemma 11.1.1 we have:
hσ(R), σ(R)i = w2 (P ) (mod 2)
and the corollary follows.
Now we are ready with the promised refinement of Theorem 1.1.1.
Theorem 11.2.4. Suppose the surface R and homomorphism θ satisfy
the hypothesis of Theorem 1.1.1. Suppose that D is a nonnegative divisor on R
such that w2 (θ) = d (mod 2), where d = deg(D). Then there exists a complex
projective structure on R that has the monodromy θ and branching divisor
equivalent to D.
Proof. The proof is a straightforward generalization of the proof of Theorem 1.1.1. Let P denote the S2 -bundle over the surface R associated with the
homomorphism θ. We first construct a decomposition of the surface R into a
union of pairs of pants so that the restriction of θ to the fundamental group of
each pair of pants is a Schottky representation. We use these representations
684
DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
to build a complex projective structure on a pants configuration. But there is
a “topological” Z/2-obstruction to forming the final handle. This obstruction
is a Dehn twist along a compressing loop. Suppose first that w2 (P ) = 0. If
the obstruction is nontrivial, then we can still construct a projective structure for the pants configuration which has exactly one branch point of order 1
and the monodromy θ. However the existence of such a structure contradicts
Corollary 11.2.2. Thus the “topological” obstruction to the existence of an
unbranched structure was trivial to begin with. In parallel, we conclude that
if w2 (P ) 6= 0, then the pants configuration admits a branched structure with
a single branch point of order 1. Now consider the general case assuming that
w2 (P ) = 0. By adding to the pants configuration (for example to a single
pants in the configuration) branch points equivalent to the divisor D, we do
not change the “topological” Z/2-obstruction to completing the construction.
Since deg(D) = 0 (mod 2), adding the branch points has the effect of twisting
one of the boundary curves an even number of times. Hence for the resulting branched pants configuration there is no obstruction to completing it to a
closed surface. The construction in the case w2 (P ) 6= 0 is similar.
11.3. The algebro-geometric interpretation. Let R be a closed Riemann
surface R of genus g ≥ 2. In this section we shall consider holomorphic vector
bundles W over R such that rank(W ) = 2 and det(W ) = 1 (i.e. the determinant bundle is trivial). Let V ∗ (R) denote the collection of holomorphic vector
bundles W over R such that W admits a holomorphic flat connection. According to Weil’s theorem (see [At], [Gu2], [W]), elements of V ∗ (R) can be
characterized intrinsically as follows:
Suppose that W = ⊕j Wj is the holomorphic direct sum decomposition
of W into (holomorphically) indecomposable vector bundles. Then the bundle
W admits a holomorphic flat connection if and only if deg(det(Wj )) = 0 for
all j.
Let
F ∗ (R) := {(ξ, ∇) : ξ ∈ V ∗ (R), ∇ is a holomorphic flat connection on ξ}
be the space of local systems on R. We have the Riemann-Hilbert correspondence:
∗
RHR
: F ∗ (R) → Y (π1 (R), SL(2, C)) := Hom(π1 (R), SL(2, C))/SL(2, C)
given by the conjugacy class of the monodromy of the flat connection ∇. It is
∗ is bijective (since every flat bundle over R has
clear that the mapping RHR
a canonical complex structure). The space Y (π1 (R), SL(2, C)) has a natural
∗ is a homeomor(non-Hausdorff) topology, we topologize F ∗ (R) so that RHR
phism.
We also have the natural projection
∗
∗
: F ∗ (R) −→ V ∗ (R), πR
(ξ, ∇) := ξ.
πR
MONODROMY GROUPS
685
Recall that each holomorphic vector bundle W has the degree of instability
u(W ) defined as follows:
u(W ) = d is the maximal number such that W contains a holomorphic
line subbundle L ⊂ W such that deg(L) = d.
In general, u(W ) = d − deg(det(W ).
For all bundles W ∈ V ∗ (R),
−g ≤ u(W ) ≤ g − 1
(see for instance [Gu2]), and stable (resp. semistable) bundles W are defined by
the condition u(W ) < 0 (resp. u(W ) ≤ 0). Stable and semistable bundles and
their moduli spaces have been extensively studied by algebraic geometers since
the seminal paper of Narasimhan and Seshadri [N-S]. In contrast, our main
objects are maximally unstable bundles W which are defined by the condition
u(W ) = g − 1. Gunning [Gu1] proves that projectivizations of all maximally
unstable bundles over R are holomorphically isomorphic to each other. We let
MR denote the corresponding projective bundle over R. It gives rise to a finite
subset MR∗ of VR∗ that consists of 22g vector bundles that can be described
as follows. Let K denote the canonical bundle on R. Choose a holomorphic
line bundle L on R such that L2 = K. Then deg(L) = g − 1. There are 22g
characters χ : π1 (R) → {±1} ⊂ C. Each character gives rise to a holomorphic
line bundle over R which√we shall denote by the same letter χ. Then the
2g
collection of square roots
√ K of the bundle K consists of 2 bundles χ ⊗ L.
For each Λ = χ ⊗ L ∈ K there is a unique holomorphically indecomposable
bundle W = Wχ for which there is a short exact sequence
1 → Λ → W → Λ−1 → 1
of holomorphic morphisms of holomorphic bundles. Notice that Wχ = χ ⊗ W1
where 1 : π1 (R) → {±1} is the trivial homomorphism. Then
MR∗ = {Wχ , χ : π1 (R) → {±1}}.
Also in [Gu1], Gunning establishes the basic relation between maximally
unstable bundles and complex projective structures on the surface R. He
proves that
∗
∗ −1
((πR
) (MR∗ )) ⊂ Y (π1 (R), SL(2, C))
RHR
consists of (conjugacy classes of) linear monodromy representations of complex projective structures on the Riemann surface R. The relation between
(branched) complex projective structures and instability of holomorphic vector bundles is further explored in [Man1, 2, 3].
The results of the previous two sections imply the following:
686
DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
Corollary 11.3.1. Suppose that θ : π1 (R) → PSL(2, C) is the monodromy representation of a branched projective structure with branching divisor D. Let P → R denote the associated S2 -bundle over R which is the
projectivization of a holomorphic vector bundle V → R. Then u(V ) ≥ g − 1 +
[deg(det(V )) − deg(D)]/2.
Proof. The developing map of the projective structure defines a section σ
of the bundle P , let L ⊂ V be the corresponding line subbundle. Then Lemma
11.1.1 and Proposition 11.2.2 imply that
deg(L) = g − 1 + [deg(det(V )) − deg(D)]/2.
From now on it will be convenient to projectivize all vector bundles, connections and representations. Let
Y (π1 (R), PSL(2, C)) := {p(ρ), ρ ∈ Hom(π1 (R), SL(2, C))}/PSL(2, C) ⊂ Vg ,
where p(ρ) is the projectivization of ρ. Denote the spaces of projectivized
holomorphic bundles and local systems over R by V (R) and F (R) respectively.
Let RHR : F (R) → Y (π1 (R), PSL(2, C)) denote the induced Riemann-Hilbert
correspondence. Similarly define the projection πR by projectivizing the
∗.
mapping πR
Our next step is to allow the complex structure on the surface R to vary.
We let S be the oriented smooth surface underlying R. Let (S) denote the
Teichmüller space of S. Consider the spaces
T
Vtop (S) :=
[
T
V (R),
R∈ (S)
Ftop (S) :=
[
T
F (R),
R∈ (S)
and mappings,
π : Ftop (S) → Vtop (S) , RH : Ftop (S) → Y (π1 (S), PSL(2, C)),
whose restrictions to the fibers F (R) are πR : F (R) → V (R) and RHR .
Remark 11.3.2. The space Ftop (S) is naturally identified with the product
T
Ftop (S) = (S) × Y (π1 (S), PSL(2, C)).
T
T
The projection Ftop (S) → (S) which maps F (R, φ) to (R, φ) ∈ (S) is the
projection of Ftop (S) to the first factor of the product decomposition.
Indeed, suppose (R, φ) ∈ (S) is a marked Riemann surface with the
marking φ : π1 (S) → π1 (R) (which is an isomorphism defined up to an inner
automorphism). Then φ indices an natural isomorphism
T
Y (π1 (R), PSL(2, C)) → Y (π1 (S), PSL(2, C))
687
MONODROMY GROUPS
M
given by precomposition of representations with φ. Note that we have to work
with the Teichmüller space of S rather than with the moduli space
(S),
otherwise the natural projection to (S) would be a nontrivial fibration (in
the orbifold sense).
M
The projection
T
Π : Vtop (S) → (S), Π : V (R) → R
has a section
µ : R 7→ MR ∈ V (R) ⊂ Vtop (S),
where MR is the projectivization of maximally unstable vector bundles over
R. Let
Yne (π1 (S), PSL(2, C)) ⊂ Vg0
denote the collection of conjugacy classes of all projectivized nonelementary
representations into SL(2, C). We summarize this in the diagram below:
Tx(S)
µ
−→
Vtop (S)
π −1 (µ( (S)))
⊂
Ftop (S)
Yne (π1 (S), PSL(2, C))
⊂

Π ◦ π

T


RH
y
x

π



RH
y
Y (π1 (S), PSL(2, C)) .
T
In view of [Gu1], the image RH(π −1 (µ( (S)))) consists of (projective) monodromy representations of complex projective structures on the surface S.
On the other hand, each holomorphic bundle in MR∗ is maximally unstable. Let Vρ be a maximally unstable bundle associated with a representation
ρ : π1 (R) → SL(2, C). Thus, for all characters χ : π1 (R) → {±1}, the bundles
χ⊗Vρ = Vχ·ρ are also maximally unstable. The inverse image of the subvariety
π −1 (MR ) in Y (π1 (R), SL(2, C)) has 22g components. Each component consists
of holomorphically isomorphic vector bundles over R, but members of distinct
components are not holomorphically isomorphic to each other.
Therefore, by applying Theorem 1.1.1, we obtain,
T
Theorem 11.3.3. The map RH sends π −1 (µ( (S))) onto Yne (π1 (S),
PSL(2, C)). In other words, let ρ ∈ Y (π1 (S), SL(2, C)) be a nonelementary
representation. It is the monodromy of a holomorphic flat connection on a
maximally unstable holomorphic vector bundle over a Riemann surface R; R
is diffeomorphic to S via an orientation-preserving diffeomorphism.
688
DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
11.4. Proper embeddings in the representation variety. In this section we
will give a detailed proof of the “divergence” theorem. It was first suggested
by Hejhal in [He1] that such theorem could be true. This theorem shows that
on a fixed Riemann surface, if any sequence of quadratic differentials diverge,
so must the conjugacy classes of corresponding monodromy representations. A
brief outline of the proof was given in [Ka, §7.2].2
As before, R denotes a closed Riemann surface of genus exceeding one
and Q(R) its space of holomorphic quadratic differentials. Let hol denote the
map that sends each φ ∈ Q(R) to the monodromy homomorphism determined
by the corresponding Schwarzian equation S(f ) = φ. By Theorem 1.1.1, the
image lies in the component of the representation variety Vg containing the
identity (cf., §1.5). That is,
hol : Q(R) → Y (π1 (R), PSL(2, C)).
Theorem 11.4.1 (Divergence Theorem). The map hol is proper.
Proof. Let Z̃ ⊂ Hom(π1 (R), SL(2, C)) denote the preimage of Z, where Z
is the image of hol.
Our first goal is to show that Z̃ is a properly embedded complex analytic
subvariety in Hom(π1 (R), SL(2, C)). Indeed, if ρ : π1 (R) → SL(2, C) is any
representation, the associated vector bundle Vρ → R is maximally unstable if
and only if ρ ∈ Z̃. Equivalently,
√
ρ ∈ Z̃ ⇐⇒ H 0 (R, L∗ ⊗ Vρ ) 6= 0 for some L ∈ K.
√
The set K is finite. Thus, by the upper semicontinuity theorem for cohomology (see [B-S]), the subset Z̃ is closed and is equal to a finite union of disjoint
complex analytic subvarieties XL properly
√ embedded in Hom(π1 (R), SL(2, C))
(these subvarieties are indexed by L ∈ K).
Recall that Z̃ is contained in the open subset Homne (π1 (R), SL(2, C)) of
nonelementary representations, i.e. those whose projectivizations are nonelementary. The group SL(2, C) acts on Homne (π1 (R), SL(2, C)) by conjugation
and the quotient is Yne (π1 (R), SL(2, C)). Hence the projection
Homne (π1 (R), SL(2, C)) → Yne (π1 (R), SL(2, C))
is a principal SL(2, C)-bundle. Since Z̃ is invariant under this action, the
projection Z ∗ of Z̃ to Yne (π1 (R), SL(2, C)) is again a closed properly embedded
complex
analytic subvariety. It consists of 22g components indexed by elements
√
of K.
The restriction of the projection
p : Yne (π1 (R), SL(2, C)) → Yne (π1 (R), PSL(2, C))
that the discussion in [Ka, §7.2] does not distinguish linear and projective monodromy
representations.
2 Note
MONODROMY GROUPS
689
to each component of Z ∗ is a bijection onto hol(Q(R)). Now p(Z ∗ ) = Z is
closed, since p is a finite covering. It is disjoint from the collection of conjugacy
classes of elementary representations because all elementary representations
correspond to semistable bundles over R. Consequently we can restrict our
study to the smooth (Hausdorff) manifold Yne (π1 (R), PSL(2, C)).
According to [Gu2], the partition of Yne (π1 (R), SL(2, C)) into holomorphic
equivalence classes is a smooth foliation. The components of Z ∗ are leaves of
this foliation; hence they are complex submanifolds in Yne (π1 (R), SL(2, C)).
This implies that Z ⊂ Yne (π1 (R), PSL(2, C)) is a properly embedded complex
submanifold. On the other hand, the mapping hol : Q(R) → Z is a continuous bijection, hence a homeomorphism. Therefore hol : Q(R) → Z ⊂
Yne (π1 (R), PSL(2, C)) is proper. Hence the composition of
hol : Q(R) → Yne (π1 (R), PSL(2, C))
with the inclusion
Yne (π1 (R), PSL(2, C)) ,→ Y (π1 (R), PSL(2, C))
is a proper map Q(R) → Y (π1 (R), PSL(2, C)).
Remark 11.4.2. The above proof shows that elementary representations
cannot be limits of sequences from hol(Q(R)). It was proven [Ka] only that
the mapping hol : Q(R) → Yne (π1 (R), PSL(2, C)) is proper. Tanigawa [Tani]
recently gave a nice geometric proof of this statement in contrast to algebrogeometric proof presented here and in [Ka]. However Tanigawa’s arguments
do not seem to prove that Z = hol(Q(R)) is closed in Y (π1 (R), PSL(2, C)),
only in the submanifold corresponding to nonelementary representations. See
also §12.4.
11.5. An analogue of Hejhal ’s holonomy theorem for branched projective
structures. The nonelementary representation variety Vg0 has two components
[Go2]. These correspond to the representations that lift to SL(2, C), and those
that do not. Each of these has dimension 6g−6. By a singly branched projective
structure we mean one that has exactly one branch point and that is of order
two. In the next section we will show that the monodromy of each singly
branched projective structure is a nonelementary representation but we will
use this fact in this section.
Let R be a closed Riemann surface of genus g ≥ 2 and p ∈ R a given
point. We will first parameterize singly branched structures on R with branch
point at the designated point p. Let D be the divisor of p and QD (R) the
space of holomorphic quadratic differentials on R which have at most a simple
pole at p.
Recall from §1.4, equation (6), that the meromorphic quadratic differential
φ0 generates a singly branched complex projective structure if its Laurent
690
DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
expansion at the chosen branch point p has the form
φ0 = −3/z 2 + b/z + α0 + α1 z + · · · ,
where b2 + 2α0 = 0,
(here and below we choose local coordinates so that p is identified with zero).
The side condition comes from the requirement that the solution of the
Schwarzian equation has no logarithmic term.
We note that there exists such a differential φ0 . First of all the RiemannRoch theorem implies there is a quadratic differential with a double pole at
any point p. Secondly it also implies that there is an abelian differential ω
which does not vanish at p. The holomorphic differential ω 2 can be employed
to insure that the side condition is satisfied ([Man1]). Fix one such quadratic
differential φ0 .
There is a meromorphic quadratic differential with a single pole at p with
the Laurent expansion
ψ0 = 1/z + a0 + d1 z + · · · .
Adding ω 2 , which does not vanish at p, to ψ0 if necessary, we may assume
that a0 + b 6= 0.
Let ψi , 1 ≤ i ≤ 3g −3 be a basis of the holomorphic quadratic differentials
on R. Then ψi , 0 ≤ i ≤ 3g − 3 is a basis of the space QD (R).
Let ai be the constant term in the Laurent expansion of ψi at p. Not all
ai can vanish.
P3g−3
The vector space QD (R) consists of the differentials ψ =
i=0 ci ψi .
When is φ0 + ψ an admissible quadratic differential in the sense of §1.4, equation (6)? The answer is when u2 + 2bu + 2v = 0, where v is the constant term
in in the Laurent expansion of ψ at p, and u is its residue.
P
The constant term in ψ is v = 3g−3
i=0 ci ai . The residue term is just c0 .
Hence the condition reads
(11)
c20
+ 2bc0 + 2
3g−3
X
ci ai = 0.
i=0
Recall that a0 + b 6= 0, thus the implicit function theorem implies that the
collection of vectors ~c = (c0 , c1 , . . . , c3g−3 ) satisfying the above equation is a
complex manifold of dimension 3g − 3 provided that the norm |~c| is sufficiently
small. (Actually it suffices to require that only |c0 | is sufficiently small.)
Consequently we can choose a small neighborhood U of φ0 in the affine
space of meromorphic quadratic differentials φ0 + QD (R) with the following
property.
P
The collection of differentials φ0 + 3g−3
i=0 ci ψi ∈ U satisfying (11) forms a
(3g − 3)-dimensional complex manifold ∆ containing φ0 .
Let Bg denote the holomorphic variety which consists of singly branched
complex projective structures on closed Riemann surfaces S of genus g ≥ 2.
691
MONODROMY GROUPS
T
Let (S − {q}) denote the Teichmüller space of surfaces S with one marked
point. There is a holomorphic mapping ν : Bg → (S − {q}) whose fiber
over a marked Riemann surface R with a marked point p is the space I(R, D)
of singly branched complex projective structures with the underlying complex
structure R and branching at D = p. It follows from the above discussion that
Bg is a holomorphic variety of generic dimension 6g − 5: the Teichmüller space
of once punctured surfaces (S − {q}) has complex dimension 3g − 2 and the
fiber of ν has complex dimension 3g − 3. There is an open and dense subset of
Bg which is a complex manifold of dimension 6g − 5. We will use the notation
(S, q, ϕ) for elements of Bg , where S denotes the marked Riemann surface, q
the branch point and ϕ the meromorphic quadratic differential.
We will need the following explicit description of the space Bg . Choose
a point R as the “origin” in (S) and write R = H2 /G where H2 is the unit
disk {|z| < 1} and G is a fuchsian group acting on H2 . In the “Bers’ slice”
model, Teichmüller space (S) is identified with that subset of the space Q(R)
of holomorphic quadratic differentials on R, lifted to H2 , such that the corresponding developing map hτ : H2 → S2 , τ ∈ Q(R), is a univalent holomorphic
mapping with homeomorphic extension to {|z| = 1}. Thus h = hτ solves
the Schwarzian equation for τ ; we will normalize it by the requirement that
h(0) = 0, h0 (0) = 1. Let ρτ : G → Gτ denote the corresponding monodromy
representation. As τ → 0, Gτ converges algebraically back to G.
The image ρτ (G) = Gτ is a quasifuchsian group. Its set of discontinuity
has two components. One is Ωτ = hτ (H2 ). The other Ωτ represents the marked
Riemann surface Rτ := Ωτ /Gτ ∈ (S). The homotopy marking of this point
in (S) is given by the isomorphism ρτ : G = π1 (R) → Gτ = π1 (Rτ ). If we
mark a point p ∈ Rτ we get an element of (S − {q}).
Any given compact subset of Ω0 belongs to Ωτ for τ sufficiently close to
0; likewise any neighborhood of the closure of Ω0 contains Ωτ for τ sufficiently
close to 0. Here Ω0 = {z : |z| > 1} ∪ ∞.
T
T
T
T
T
T
T
Lemma 11.5.1. There is a locally defined holomorphic map P : Bg → S2
that “records” the position of the branch points.
Proof. We construct P in a small neighborhood ∆ of a given point (R, q, ϕ)
∈ Bg , where ϕ is a quadratic differential on R, the surface R is the “origin” in
the Teichmüller space and q ∈ R is the branch point. We will denote points
σ ∈ ∆ by (Rτ , qσ , ϕσ ) where τ = τ (σ) ∈ (S) and qσ ∈ Rτ is the branchpoint. If τ = 0, then σ represents a change of branch point from q to qσ on
R itself. The point σ = 0 is (R, q, ϕ). Let f0 : Ω0 → S2 denote the (as yet
unnormalized) developing map of (R, q, ϕ) and θ : G = π1 (R) → PSL(2, C) the
associated nonelementary monodromy representation (here we are applying
Theorem 11.6.1 that will be proven in the next section).
T
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DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
Let fσ : Ωτ → S2 be the associated holomorphic developing map. We will
show in the next paragraph how to fix a consistent normalization for fσ given
f0 so that the restrictions of fσ to compact domains in Ω0 depend holomorphically on σ. Each developing mapping fσ corresponds to the monodromy
representation
θσ0 := θσ ◦ ρτ : G → Gτ → Hσ ⊂ PSL(2, C).
At the origin, θ00 = θ0 .
Consider the projection Hom(π1 (R), PSL(2, C)) → Vg . We will construct
a local cross section Veg near θ00 as follows. We know from Part A that we can
find in H0 three loxodromic elements h1 = θ00 (g1 ), h2 = θ00 (g2 ), h3 = θ00 (g3 ) with
distinct attracting fixed points a1 , a2 , a3 , where g1 , g2 , g3 ∈ G. Normalize each
developing mapping fσ so that the attracting fixed point of θσ0 (gj ) remains
aj , j = 1, 2, 3. This can be done for all σ ∈ ∆ if ∆ is sufficiently small, i.e.,
if the attracting fixed points remain distinct and the elements θσ0 (gj ) remain
loxodromic. Thus in ∆ we have a holomorphic lift
g : Bg → Veg ⊂ Hom(G, PSL(2, C)).
hol
Now given a lift q ∗ ∈ Ω0 of q ∈ R, there is, in the set of lifts of qσ to Ωσ , a
closest (in the spherical metric) point qσ∗ to q ∗ . Define
P : (Rτ , qσ , ϕσ ) 7→ fσ (qσ∗ ) ∈ S2 .
It is clear that the mapping P is holomorphic provided that ∆ is so small that
the point qσ∗ is unique.
Thus, by the previous lemma we have a locally defined holomorphic map
µ =
(P, hol) : Bg → S2 × Vg
µ̃ =
g : Bg → S2 × Veg .
(P, hol)
and its lift
We are now ready to state our theorem.
Theorem 11.5.2. The holonomy map hol : Bg → Vg is locally a topological fiber bundle with fiber of complex dimension one.
Remark 11.5.3. The fibers reflect the choice of branch point. This result
should generalize to the space of D-branched projective structures where D is
a fixed (topological) branching divisor, provided we consider structures with
nonelementary monodromy.
MONODROMY GROUPS
693
Proof. In Lemmas 11.5.5 and 11.5.4 below we will prove that µ is injective
and an open map. Hence µ is a local homeomorphism. Since S2 × Vg0 is a
complex manifold of dimension (6g−5) we can therefore use µ to locally identify
Bg with the product S2 × Vg so that hol is identified with the projection to the
second factor.
Lemma 11.5.4. Let X be a holomorphic variety of generic complex dimension n (i.e. there is an open dense subset U ⊂ X which is a complex
manifold of dimension n). Let f : X → M be a locally injective holomorphic
mapping, where M is a complex manifold of dimension n. Then f is open.
Proof. Since this is a local question it suffices to consider the germ of X
at a point x ∈ X and the germ of f at x. Since f is locally injective, the germ
of the mapping f at x is “finite” in the terminology of [Gu4, p. 56].
Suppose that the germ of f at x is not onto. Apply [Gu4, Corollary 9]: it
follows that there exists a nonzero germ of a holomorphic function h on M at
m = f (x) such that h ◦ f = 0. The germ at m of the zero level set {h = 0} of h
is a holomorphic subvariety of dimension strictly less than n, by the uniqueness
principle of holomorphic functions. Thus the germ of the image f (X) at m has
generic dimension less than n. However f (X) is generically a manifold, hence
f (U ) has dimension less than n, a contradiction to invariance of domain for
manifolds.
Lemma 11.5.5. The mapping µ is locally injective.
Proof. It suffices to show that two nearby branched structures with the
same monodromy representation are identical provided that the images of their
branch points under P are the same. Our proof is analogous to that of [He1,
Theorem 1]. It clearly enough to show local injectivity of the holomorphic lift
g : Bg → S2 × Veg .
µ̃ = (P, hol)
We consider the points σ = (Rτ , qσ , ϕσ ) of a small neighborhood ∆ of the point
(R, q, ϕ) ∈ Bg .
Let Fσ ⊂ Ωτ denote the (closed) Dirichlet fundamental domain for Gτ in
the hyperbolic metric on Ωτ and with center qσ∗ ; τ = τ (σ). Let F0∗ be a small
open neighborhood of F0 , and take ∆ so small that Fσ ⊂ F0∗ for all σ ∈ ∆.
We may also assume that the orbit Gτ (qσ∗ ) meets the closure of F0∗ only at qσ∗ .
We again use the developing mappings fσ : Ωτ → S2 . Decreasing ∆ even
more if necessary, we may assume that:
a) For each σ ∈ ∆ there is an open neighborhood Fσ∗ of Fσ such that for
any pair of points σ, δ ∈ ∆ we have Fδ ⊂ Fσ∗ , and
b) Given small ε > 0, there is a disk V ⊂ F0 about q ∗ of radius 2ε with
the following property. To any z ∈ F0∗ \ V , and to any pair of points σ, δ ∈ ∆,
694
DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
corresponds a unique point zσ,δ ∈ F0∗ such that
fδ (zσ,δ ) = fσ (z),
and d(zσ,δ , z) < ε.
Here d(·, ·) is the spherical metric.
The “membranes” {fσ (Fσ )} over S2 serve as “fundamental domains” for
the image groups Hσ = θσ0 (G).
Suppose µ̃ is not injective in any neighborhood ∆. Then for arbitrarily
small ∆ there exist σ 6= δ ∈ ∆ so that P (σ) = P (δ) and the normalized
monodromy representations are identical, i.e.,
(θσ0 : G → Hσ ) ≡ (θδ0 : G → Hδ ).
We claim that there is a branch F of fδ−1 ◦ fσ which is a conformal homeomorphism of the fundamental domain Fσ onto a new fundamental domain
Fδ0 for Gτ (δ) in Ωτ (δ) . Such a map F would necessarily be equivariant in
the sense that if z, g(z) ∈ Fσ , g ∈ Gτ (σ) , then F (z), F (g(z)) ∈ Fδ0 and
−1
F (g(z)) = ρτ (δ) ◦ ρ−1
τ (σ) (g)F (z). Here ξ := ρτ (δ) ◦ ρτ (σ) : Gτ (σ) → Gτ (δ) is
the isomorphism which factors through G.
Indeed, for z ∈
/ V define F (z) := zσ,δ . It is clear F is a univalent holomorphic mapping. Furthermore F |Fσ \ V extends over V to a conformal mapping
because both fσ and fδ are 2-fold branched coverings near q ∗ ∈ Ω0 with the
same critical value
fσ (qσ∗ ) = P (σ) = P (δ) = fδ (qδ∗ ).
The mapping F projects to a conformal map of Rτ (σ) = Ωτ (σ) /Gτ (σ) onto
Rτ (δ) = Ωτ (δ) /Gτ (δ) . Correspondingly F extends to a conformal mapping
F : Ωτ (σ) → Ωτ (δ) that induces the isomorphism ξ : Gτ (σ) → Gτ (δ) .
The map hτ (δ) ◦ h−1
τ (σ) is a conformal map of Ωτ (σ) onto Ωτ (δ) which also
induces the isomorphism ξ. The two conformal mappings have continuous
extensions to the limit set which are necessarily identical. Since the limit set is
a quasicircle they are the restrictions of a Möbius transformation. In particular
F is a Möbius transformation and σ = δ, a contradiction.
The following is a direct consequence of Theorem 11.5.2.
Corollary 11.5.6. Let σ = (R, p, ϕ) be a singly branched projective
structure. Let ∆ ⊂ Bg be a sufficiently small neighborhood of σ in the space of
singly branched structures δ on R “with the same image of the branch point”
P (δ) as σ. Suppose the sequence of normalized representations θi : π1 (R) →
PSL(2, C) converges algebraically to the normalized monodromy representation
θ associated with σ. Then for all large i, θi is associated with a unique σi ∈ ∆.
11.6. Monodromy of singly branched projective structures. In this section
we will prove facts that have been announced in §1.6.
MONODROMY GROUPS
695
Theorem 11.6.1. Suppose that R is a closed Riemann surface of genus
g ≥ 2, θ : π1 (R) → PSL(2, C) is the monodromy representation of a singly
branched complex projective structure τ on R. Then Γ = θ(π1 (R)) is a nonelementary subgroup of PSL(2, C).
Proof. Since τ has exactly one branch point and the order of this branch
point is 2, the representation θ has nonzero 2nd Stiefel-Whitney class. In
particular, θ cannot be lifted to a representation π1 (R) → SL(2, C). Suppose
that the group Γ = θ(π1 (R)) is elementary. There are three cases:
(a) The group Γ has a fixed point z ∈ S2 . Without loss of generality
we can assume that z = ∞, thus Γ is contained in the group A of complex
affine transformations of C. The inclusion A ,→ PSL(2, C) admits a 1-1 lift
A ,→ SL(2, C)
µ
¶
a ba−1
2
.
a z + b 7→
0 a−1
Therefore θ lifts to a representation θ∗ : π1 (R) → SL(2, C), which contradicts
the assumption that θ has nonzero 2nd Stiefel-Whitney class.
(b) Suppose that Γ is conjugate into the subgroup P U (2) ⊂ PSL(2, C).
Let R̃ → R be a 2-fold covering over R. Thus 2(g − 1) = g̃ − 1, where g̃ denotes
the genus of R̃. The complex projective structure τ on R defines a complex
projective structure τ̃ on R̃ with two branch points of order two. Suppose that
Γ ⊂ P U (2); then θ(π1 (R̃)) ⊂ P U (2) as well. The representation θ|π1 (R̃) lifts
to a linear representation
θ∗ : π1 (R̃) → SU (2) ⊂ SL(2, C).
Consider the flat vector bundle V of the rank 2 over the surface R̃ associated
with the action θ∗ of π1 (R̃) on C2 . Clearly det(V ) = 1. The developing map
of the branched complex projective structure τ̃ defines a section
σ : R̃ → P (V ).
According to Proposition 11.2.2, the self-intersection number σ 2 of the surface
σ(R̃) in P (V ) equals (2 − 2g̃) + 2, since the structure τ̃ has exactly two branch
points of the order 2.
It follows from Lemma 11.1.1 that the section σ gives rise to a line subbundle L ⊂ V such that
deg(L) = (g̃ − 1) − 1 = 2g − 3 > 0.
We conclude that u(V ) > 0 and the bundle V is unstable. On the other hand,
every flat bundle over R̃ with unitary monodromy group is semistable (see
for instance [N-S]). This contradiction shows that Γ cannot be contained in
P U (2).
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DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
(c) Consider the case that the group θ(π1 (R)) has an invariant pair of
points in S2 . (This does not imply that θ can be lifted to SL(2, C).) We argue
as in Case (b). There is a 2-fold covering R̃ → R such that the group θ(π1 (R̃))
has a pair of fixed points in S2 . Therefore the induced complex projective
structure on R̃ has two branch points and the monodromy group θ(π1 (R̃)) has
a lift θ∗ (π1 (R̃)) to a subgroup of SL(2, C) conjugate to the group of diagonal
matrices. Let V denote the holomorphic vector bundle associated with the
representation θ∗ : π1 (R̃) → SL(2, C). The representation θ∗ splits as the
direct sum of representations. Hence the bundle V is decomposable (into the
direct sum of two line bundles of degree zero), which implies that u(V ) = 0.
On the other hand, the developing map of the branched complex projective
structure on R̃ defines a section σ : R → P (V ) with the self-intersection
number
(2 − 2g̃) + 2 < 0,
where g̃ denotes the genus of R̃. Hence u(V ) > 0 which contradicts u(V ) = 0.
Suppose that τ is a branched complex projective structure on the closed
Riemann surface R of genus at least two. We identify the universal cover of R
with the hyperbolic plane H2 . Let f : H2 → S2 be the developing map of τ and
Γ = θ(π1 (R)) be the holonomy group. We say that τ is a branched hyperbolic
structure if τ has at least one branch point and the image of f is a round disk
in S2 . This definition is motivated by the fact that in such case Γ preserves
the hyperbolic metric ds2 in f (H2 ). The pull back of ds2 from f (H2 ) to R is
a hyperbolic metric on R which has singular points at the branch points zj of
τ ; the total angle around zj is 2πkj , where kj is the order of zj .
Later we will show by example why the following result is false if we do not
exclude branched hyperbolic structures. This too has been announced in §1.6.
Corollary 11.6.2. Suppose that either the complex projective structure
(f, θ) is unbranched, or is singly branched but is not a branched hyperbolic
structure (i.e. f (H2 ) is not a round disk ). Then the following statements are
equivalent:
(i) f (H2 ) 6= S2 ;
(ii)
(iii)
H2 → f (H2 ) is a (possibly branched ) cover ;
Γ acts discontinuously on f (H2 ).
Proof. The unbranched case is classical (see §1.6). Consider then the
branched case. By Theorem 11.6.1, Γ = θ(π1 (R)) is nonelementary. The limit
set Λ(Γ) is the smallest Γ-invariant closed nonempty subset of S2 . Since Γ
MONODROMY GROUPS
697
is nonelementary, Λ(Γ) is the closure of the set of fixed points of loxodromic
elements of Γ. It follows that the Γ-orbit of any open set containing a limit
point is S2 . Suppose that Γ ⊂ PSL(2, C) is nondiscrete. Let Γ̄ be the closure of
Γ in PSL(2, C). Since Γ is nonelementary it follows that Γ̄ is either PSL(2, C)
or it preserves a round circle C ⊂ S2 and Λ(Γ) = C [Gr]. If the latter case
occurred, f (H2 ) would be one of the two round disks in S2 bounded by C. It
would follow that τ is a branched hyperbolic structure in contradiction to our
assumption. If Γ̄ = PSL(2, C) then f (H2 ) is contained in Λ(Γ) = S2 which
implies that f (H2 ) = S2 .
We conclude that if (i) holds then Γ is a discrete subgroup of PSL(2, C)
and f (H2 ) is contained in the discontinuity domain Ω(Γ) = S2 \ Λ(Γ). Hence
(i) ⇒ (iii). Clearly, (iii) ⇒ (i).
The implication (ii) ⇒ (i) is immediate. Conversely if (iii) holds, f (H2 )
must be contained in a component ∆ of the domain of discontinuity of Γ. Since
f (H2 ) is connected and Γ-invariant it follows that ∆ is also Γ-invariant. Hence
f projects to a holomorphic map fˆ : R → fˆ(R) ⊂ Σ = ∆/Γ. Since fˆ(R)
is a compact subsurface without boundary in Σ we conclude that fˆ(R) = Σ
and Σ is a closed surface. Any nonconstant holomorphic surjective mapping
between closed Riemann surfaces is necessarily a covering, possibly branched.
Consequently f itself is a possibly branched covering map.
We will now construct an example of a singly branched hyperbolic structure on a surface R of genus two which has nondiscrete holonomy in PSL(2, R).
Start with a regular hyperbolic octagon X ⊂ H2 with vertex angles π/2
−1
−1
in positive order around
(cf. [Tan]). Label the edges b−1
1 , a1 , b1 , a1 , . . . a2
X. Identify the edges by corresponding isometries A1 , B1 , A2 , B2 to obtain a
Riemann surface of genus two such that H2 is a two sheeted cover branched
over one point on R. Let σ denote the line segment from the left end point of
−1
b−1
1 to the right end point of a1 Then
−1
−1 −1
A1 B1 A−1
1 B1 = E = A2 B2 A2 B2
where E is a elliptic transformation of order two fixing the midpoint of σ. Let
γ denote the branched projective structure on R with the holonomy group
Γ = hA1 , B1 , A2 , B2 i. The quotient orbifold H2 /Γ is a torus with one cone
point of order two. Clearly the holonomy θ : π1 (R) → Γ is not injective (cf.
[Go1]). According to Theorem 1.1.1, θ does not lift to SL(2, R).
Next, we will show there exists a hyperbolic structure with exactly one
branch point of order two and a nondiscrete holonomy group. Take the example
above of a branched structure γ. The representation variety
Hom(Γ, PSL(2, R))/PSL(2, R)
is 2-dimensional and the representation variety
Hom(π1 (R), PSL(2, R))/PSL(2, R)
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DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
is 6-dimensional. Therefore we can find a real-analytic curve of nonelementary
representations θt : π1 (R) → PSL(2, R), θ0 = θ, t ∈ [0, 1], which do not factor
through θ : π1 (R) → Γ. The fact that θt is a real-analytic curve implies that
there is a dense subset J ⊂ [0, 1] so that K = ker(θt ) = ker(θs ), s, t ∈ J. Let
Γ0 := π1 (R)/K. We claim that there cannot be a sequence of t ∈ J which
converge to t = 0 such that each Γt := θt (π1 (R)) is discrete. For otherwise a
sequence of discrete nonelementary representations ρt : Γ0 → Γt , t ∈ J would
converge to ρ : Γ0 → Γ as t → 0. The limit ρ of such sequence has to be a faithful representation as well, as a consequence of [J-K]. This contradicts the fact
that ker(ρ) = ker(θ)/K 6= {1}. Thus there is an infinite sequence of nondiscrete representations θt : π1 (R) → Γt which converges to θ. In addition Γt
necessarily preserves the upper halfplane for t close to 0. By Corollary 11.5.6,
θt is the monodromy of a branched complex projective structure γt on R with
branch point likewise at z = 0.
12. Open questions about complex projective structures
In this chapter we list some unsolved problems. Some are well known in
the field, others arise from the specific analysis of this paper.
There are two general issues: the monodromy representation per se, and
the Riemann surfaces of specified type where it is induced by a particular
projective structure.
We recall from §1.5 that Qg denotes the vector bundle of quadratic differentials over Teichmüller space g and Vg0 is the subset of nonelementary representations in the representation variety Vg , modulo conjugation by PSL(2, C).
T
12.1. Existence and nonuniqueness of points in Qg with given monodromy.
Our proof exhibits two sources of nonuniqueness:
• The nonuniqueness of the pants decomposition on which the monodromy
is Schottky.
• The nonuniqueness of the pants configuration over S2 obtained from a
pants decomposition: one can use N -sheeted branched covers for arbitrarily large N .
Our Theorem 1.1.1 provides a Riemann surface for every nonelementary
representation θ. On the other hand, if we fix attention on a particular oriented
surface R, we don’t know whether all projective structures on R itself can be
obtained from the pants decomposition method. For example, can there be
a complex projective structure σ on R so that for each simple loop γ ⊂ R
with loxodromic monodromy, no element of its homotopy class is sent by the
developing map to a simple arc in S2 ?
MONODROMY GROUPS
699
For the case of representations into PSL(2, R) all projective structures can
be obtained by the pants decomposition method, see [F], [Go1], [Ga2]. However
in all three papers the proofs that the developing map is a covering over the
upper and the lower half-planes have the same gap: In general the pull-back of
a complete Riemannian metric on a manifold via a local diffeomorphism can
be incomplete. For complete proofs of the assertion about covering see [Kui,
pp. 485–486], [Kul-Pin], or [Cho-L]).
For those projective structures on R which do arise from pants decompositions, are there optimal choices for the decompositions? For example, does
the developing mapping send each pants of some decomposition directly into
the domain of discontinuity of the corresponding Schottky group?
Problem 12.1.1. Characterize and classify the nonuniqueness of projective structures with given monodromy.
In particular is it possible to get one projective structure on R from another by a specific series of “moves”?
One might ask to do this through a sequence of graftings. Yet, at least in
the case of a once-punctured torus R, a connection solely by means of a grafting
sequence is known to be impossible in general. The reason has to do with the
fact that in the Bers slice, the result of pinching R along a simple nondividing
loop γ is a B-group Γ representing the punctured torus on one side, and the
triply punctured sphere on the other. Specifically, construct two complexprojective structures on R with the monodromy G → Γ as follows. Consider
simple nondividing loops α and β on the surface R so that all the loops α, β, γ
are mutually non-homologous. Let σt , t ∈ [0, 1) denote the family of complexprojective structures on R which is being pinched along γ as t → 1. Let
grα (σt ), grβ (σt ) be the complex-projective structures on R obtained from σt via
grafting along α and β. One can show that grα (σt ), grβ (σt ) are convergent to
complex-projective structures σ10 , σ100 on R as t approaches 1. There results two
structures σ10 , σ100 with the same orientation and the same monodromy G → Γ.
However these complex projective structures are not related by grafting. The
underlying reason is that the “complex of simple loops” on the once punctured
torus R is totally disconnected.
For branched structures, there is another way of changing projective structures without changing the monodromy. This is the method of “bubbling.”
Suppose that R is a Riemann surface with a (branched) projective structure σ. Let α ⊂ R be a compact simple arc, disjoint from the singular points of
σ, which the developing map sends to simple arcs in S2 . Let a be one of these
arcs in S2 . Then split R open along α, split S2 open along a, take N copies of
the Riemann surface S2 − a and glue them to R − α with appropriate identification of boundary edges. The net result is a projective structure on the new
700
DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
“bubble-on” Riemann surface RN with the same monodromy. The projective
structure on RN has two additional branch points (at the end points of α),
both of order N .
“Bubble-off” is the inverse operation on R.
Problem 12.1.2. Suppose that σ1 , σ2 are complex -projective structures
on a surface R with the same monodromy representation. Can one pass from σ1
to σ2 using the following elementary moves: “grafting,” inverse to “grafting,”
“bubble-on,” “bubble-off ”?
12.2. Surfaces with punctures. What about surfaces with punctures where
the corresponding quadratic differentials have at most double poles? As with
compact surfaces, the dimension of the vector bundle Q(g,n) of quadratic differentials over the Teichmüller space (g,n) agrees with that of the representation
variety, if one allows arbitrary monodromy at the punctures (for an analysis
of the derivative of the monodromy map for this case see [Luo]). One can
search again for pants decompositions, provided the monodromy is not elliptic
of infinite order at a punctures. With discrete monodromy at the punctures,
one can look for representations of fundamental groups of pants to extended
forms of Schottky groups (i.e. Klein combinations of pairs of discrete cyclic
subgroups of PSL(2, C)).
Suppose the genus of R is positive. We believe that our technique in
Part A will yield a pants decomposition of R in which the restrictions of the
monodromy are onto Schottky-like groups, provided the representation around
each puncture is a discrete (cyclic) group.
T
Problem 12.2.1. Prove and /or explore the existence and nonuniqueness
of complex projective structures with given nonelementary monodromy in the
case of punctures, most importantly and most classically, punctured spheres.
12.3. Linear monodromy representations. Throughout the paper we considered Schwarzian differential equations on Riemann surfaces. Their monodromy representations are projective representations θ : π1 (R) → PSL(2, C).
One can also consider the more general case of representations into
GL(2, C). In the classical case of punctured spheres R, the dimension of the
representation variety, modulo conjugations, is identical to the dimension of
the vector bundle over (0,n) of linear equations
T
u00 + pu0 + q = 0,
where p has at most simple poles and q double poles at the punctures. Note
that we have to restrict to the representations θ∗ into GL(2, C) which map the
peripheral loops of R to unipotent elements.
MONODROMY GROUPS
701
Problem 12.3.1. Is there an analogue of Theorem 1.1.1 for punctured
spheres if one seeks a differential equation that induces a given linear representation θ∗ ?
12.4. Divergence of monodromy representations. Fix a closed Riemann
surface R of the genus g > 1 and let φn = ϕn (z)dz 2 be a sequence of quadratic
differentials on R so that ||φn || → ∞. Let [ρn ] be the sequence of conjugacy
classes of monodromy representations of φn . We know from Theorem 11.4.1
that the sequence [ρn ] cannot subconverge to the to the conjugacy class of any
representation.
Problem 12.4.1. Characterize the “limit points” of divergent sequences
of representations in the representation variety. Prove the Divergence Theorem
11.4.1 for complex projective structures on R which have a single branch point
of order 2.
One way that the representation variety Vg can be compactified is by
(projective classes of) actions of the group G = π1 (R) on metric trees. Which
actions of G on trees can appear as limits of the sequences [ρn ]? For instance,
is it true that for each sequence of quadratic differentials φn = nφ, φ 6= 0, there
is a limit ρ of the sequence ρn with the following property: ρ is an action of G
on a tree that is dual to the singular foliation on R determined by φ?
12.5. Path lifting properties of monodromy mappings. In [He1], Hejhal
proved that the natural mapping
Pg : Qg → Vg
is a local homeomorphism which fails to be a covering mapping.
Problem 12.5.1. Let γ : [0, 1] → Vg be a continuous path, γ̃ : [0, 1) → Qg
a partial lift which can not be extended to the end -point 1. Describe the asymptotic behavior of the path γ̃.
For instance is it true that γ̃ has a well-defined limit
lim γ̃(t)
t→1
in a natural (e.g. closed ball) compactification of Qg ?
12.6. Branched projective structures. As the degree of a positive divisor D
increases, it becomes easier to construct a complex projective structure with
the branching divisor D. Thus, one should be able to eliminate the assumption that the representation θ is nonelementary for sufficiently large values of
702
DANIEL GALLO, MICHAEL KAPOVICH, AND ALBERT MARDEN
deg(D). For instance, if θ is the trivial representation, then branched structures with the monodromy θ are just m-fold ramified coverings f : R → S2 .
Thus χ(R) = mχ(S2 ) − deg(D) = 2m − deg(D). The number m is at least
2, hence deg(D) ≥ 4 − χ(R) = 2g + 2. The minimal degree is realized by a
hyperelliptic ramified covering f , for which we have: deg(D) = 2g + 2.
Problem 12.6.1. Make precise and optimize the connection between
branching divisors and monodromy. Namely, compute the function
d : Hom(G, PSL(2, C)) → Z,
where d(θ) is the smallest integer for which there exists a branched complex
projective structure with branching divisor of degree d and monodromy θ.3
We proved that d(θ) = 0 for all liftable nonelementary representations θ
and d(θ) = 1 for all nonliftable nonelementary representations θ. Is it true
that d(θ) = 2g for all liftable representations θ : G → SO(3) ⊂ PSL(2, C) and
d(θ) = 2g − 1 for all nonliftable elementary representations θ : G → SO(3) ⊂
PSL(2, C), provided that the monodromy group θ(G) is dense in SO(3)? Is it
true that d(θ) ≤ 2g + 2 for any θ : G → PSL(2, C)?
Remark 12.6.2. For the flat holomorphic bundles of rank 2 over R the
corresponding question is the following: given a representation θ∗ : G → SU(2)
with dense image, find a complex structure on R so that the associated flat
C2 -bundle V over R has the degree of instability u(V ) = −1.
St. John’s University, Staten Island, NY
E-mail address: gallod@stjohns.edu
University of Utah, Salt Lake City, UT
E-mail address: kapovich@math.utah.edu
University of Minnesota, Minneapolis, MN
E-mail address: am@math.umn.edu
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CP
(Received May 20, 1997)
(Revised November 29, 1999)